Math Forum - Problem of the Week
Submissions for Taxi!
Student Short Answer Long Answer
Student 1
1.y=0.35x+1.7
2.y=$5.90 3.you can go
18 miles Extra. 21.11
miles
Student 2
1. -0.35x + y = 1.7 2.
The cost of 12-mile
ride is $5.90 3. I can go
18 miles if I have $8 to
spend
1.I used the equation y=mx+b to find out how much it costs to ride
the miles. So i got out m=0.35 b=1.7.
2.I put 12 into the equation y=mx+b, so it became y=0.35*12+1.7, i
got y=$5.90
3.I put 8 into the equation y=mx+b, so it became 8=0.35x+1.7, I got
x=18 miles.
4.I put 10 intot he equatio y=mx+b, and times 110%, because it said
tip the cab driver 10% of the cost of the ride. It's like 10=
(0.35x+1.7)*110%, i got 21.11miles
(1)
First I have to find the slope of the line which can be drawn if I
have an equation. There is a formula for finding the slope called
the slope formula.
y1 - y2
m = --------x1 - x2
It is used to find the slope between two points and I am going to
use this formula to find the slope of the line in the question.
The mile is represent on the x-axis and the dollar which it cost is
on the y-axis, because x-axis is the one that is independent and
the y-axis is dependent(the number of dollars it cost has to depend
on how many miles it runs.)
4.15-3.10
1.05
------------- = -------- or
7-4
3
0.35
So the slope of this line is 0.35.
Now, for finding the equation and with two points, I can use the
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Page 1 of 58
point-slope form to find it, which is
y-y1 = m (x-x1)
where m is the slope of the line and (x1, y1) is any point on the
line.
So now I will substitute in one of the point and the slope of the
line.
y-3.1 = 0.35(x-4)
Distribute 0.35.
y-3.1 = 0.35x-1.4
Add 3.1 to both sides
y = 0.35x + 1.7
So now I will substitute in the other point given to see if the
equation is
right.
(4.15) = 0.35(7) + 1.7
Simplify the equation
4.15 = 2.45 + 1.7
4.15 = 4.15 Yes
So y = 0.35x + 1.7 is the equation that fits the given information.
(2)
First I know that x is the mile and y is the cost. So I will have
to
substitute 12 mile in for x in order to find out the cost of 12-mile
ride.
y = 0.35(12) + 1.7
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Page 2 of 58
Simplify the equation.
y = 4.2 + 1.7
y = 5.9
So the cost of a 12-mile ride is $5.90
(3)
I know that the cost of the ride is on the y-axis, so I will
substitute 8 in for y to find out how many miles can I ride if I
have
$8.
(8) = 0.35x + 1.7
Subtract 1.7 from both sides
6.3 = 0.35x
Divide both sides by 0.35
x = 18
I will be able to go 18 miles on the taxi ride if I have $8.
Extra:
Now I have $10 to spend, but I will have to give 10% of my money to
the cab driver as a tip. So I will have to add 10% of the original
cost to the original cost. The sum of them have to equal to 10,
which
is the dollar I can offer.
1(0.35x + 1.7) + 0.1(0.35x + 1.7)= 10
Simplify the equation.
1.1(0.35x + 1.7)= 10
Distribute 1.1
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0.385x + 1.87 = 10
Subtract 1.87 from both sides
0.385x = 8.13
Divide both sides by 0.385
x = 21.11688...
Student 3
For the first problem I
got y=.35x+1.70 as the
linear equation. For
the second porblem I
got $5.90 for a 12 mile
taxi ride. For the last
problem I got 18 miles
for 8 dollars.
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Since the question said give the answer to the nearest tenth of a
mile, I can go 21.1 miles if I have $10 to spend and have to give
10%
of the cost as the tips to the cab driver.
First I wrote out my equation : y=mx+b. I knew the x= miles and y=
amount of money and m= the slope and b= the starting amount. I
needed to find the slope of the equation because I needed that to
complete my equaion. I took the 4 (as in miles) as my x and the 3.10
(as in amount of money) for my y. I did the same with the other
exapmle they gave my with 7 as my x and 4.15 as my y. Then i
subtracted 3.10 from 4.15 to get the first part of my answer to the
slope. I got 1.05. Then I subtracted 4 from 7 to get the other half
of my answer to the slope and I got 3. Then I divided 1.05 into 3
parts and got .35. Next I needed to find the b so I used the points
4,3.10. I timesed 4 by .35 because it was my slope times x. I got
1.4. Then I subrtacted that from 3.10 to get my starting amount
which was 1.70. Next I did the same equaion but put in the points
(7,4.15). It worked so I knew my equation was right. Next i used my
equation (y=.35x+1.70) to find out a 12 mile ride's amount. I put in
12 as my x becuase it's the number of miles. I first timsed 12
by .35 and got 4.20. Then I added 1.70 to that and got 5.90. So for
a 12 mile taxi ride it would cost 5.90. Then I tried to find hoe
many miles u could travel with 8 dollars. First I made my equation
and put 8 in as my y (8=.35x+1.70). Then I minused 1.70 from each
side to make them equal my equation then looked like
this :6.30=.35x. Next I divided 6.30 by.35 to get my x. I got x to
equal 18 miles. So for 8 dollars you can travel 10 miles. I then
did the equation out all together and it worked so I knew I did it
right.
Page 4 of 58
Student 4
the linear equation is :
cost = 0.35 * distance
+ 1.7 cost(12 miles) =
5.9 $ distance(8 $) = 18
miles
Student 5
1)A linear equation
would be:
y=7/20x+1.69 2)The
cost of a 12 mile ride
would be $5.89 3)If
you have $8 to use for
a taxi ride, you can go
about 18 miles.
Student 6
The equation is y =
.35x+1.7, and if you go
12 miles it will cost you
5.9 dollars. IF you only
have eight dollars you
will get 18 miles.
1)The linear equation
is y=.35x+1.7 2) A 12mile ride would cost
$5.9 3) You can travel
18 miles with $8
Student 7
x = distance
y = cost
4.15 - 3.1 y - 3.1
---------- = --------- ==> 0.35 ( x - 4 ) = y - 3.1
7-4
x-4
==> y = 0.35 x + 1.7
1) *First I found the slope which was 7/20
*then I found the y-intercept which was about 1.69
*next I put everything toghether and got the equation (y=7/20x+1.69)
2)*First I put the 12 in where the x should be
* then I multiplied 7/20 and 12 and I got 4.2
*next I added the 4.2 with the 1.69 and got my answer of $5.89.
3)*First I pluged the 8 into the spot where the y sould be.
*then I subtracted 1.69 from both sides and got 6.31.
*next I multiplied 20/7 on both sides and got my answer of about 18
miles.
First I found out what the dependent variable was. the slope, and
then the y-intercept. Onces I had the equation I plugged in the
numbers.
1) (4,3.10), (7.4.15)
y2-y1/x2-x1
4.15-3.1/7-4= 1.05/3=.35
y=mx+b
y=.35x+b
3.1=.35(4)+b
3.1=1.4+b
-1.4 -1.4
1.7=b
y=.35x+1.7
2) y=.35x+1.7
y=.35(12)+1.7
y= 4.2+1.7=5.9
y=$5.90
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Page 5 of 58
3)y=.35x+1.7
8=.35x+1.7
-1.7 -1.7
6.3=.35x
6.3/.35=.35x/.35
18=x
Student 8
#1 1.70+.35x=y #2
$5.90 #3 24.56 miles
extra 25.71
#1
1st 3.10 - 4.15 = 1.05
2nd 1.05/7-4=3 so
3rd 1.05/3 = .35
4th 1.70 witch is a # i found by trial and eror so
5th 1.70+.35x=y the equation
#2
plug the 12 miles in to xin the equeation
1.70+.35*12
.35*12=4.2
1.70+4.2= 5.90
#3
now put 8.00 into the equation for .35
1.70+x8=y
8/.35= 22.85714 so rounded up to 22.86
22.86 goes in for the x
1.70+22.86=24.56
so with $8 you can go 24.56 miles
#extra
9/.35+25.71
Student 9
1)y=.35x+1.7
2)y=$5.90 3)x=18 miles
Extra)20.9 miles
you can go 25.71 miles on a $10 and a 10% tip
1) 4.15-3.10=1.05
1.05=3 miles
1.05/3=$0.35 per mile
4.15-(.35*7)=$1.70
y=.35x+1.7
2)y=.35(12)+1.7
y=4.2+1.7
y=$5.90
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Page 6 of 58
3)8=.35x+1.7
8-1.7=.35x+1.7-1.7
6.3=.35x
6.3/.35=.35x/.35
x= 18 miles
Student 10
1. The linear equation
that fits the given
information is
y=0.35x+1.7 2. The
cost of a 12-mile ride is
$5.90 3. If you had
eight dollars you could
travel 18 miles Extra:
You can travel about
20.9 miles with $10
and leave a 10% tip
Extra) 10*.1=1
10-1=9
9=.35x+1.7
9-1.7=.35x1.7-1.7
7.3=.35x
7.3/.35=.35x/.35
20.9 miles
1. I took 4 and $3.10 and made it into an ordered pair of (4,3.1).
Then I took 7 and $4.15 and made an ordered pair of (7,4.15).
Next I took the to orderd pairs and used the formula y2-y1 divide by
x2-x1, which gave me the slope of 0.35. Then I used the slope and one
of the ordered pairs and put it in the formula y=ax+b. I found b and
it was the y-intercept, which was 1.7. So the equation looked like
y=0.35x+1.7.
2. I took my equation and put 12 miles into the x positoin and then
figured out what y was and that was my answer, which was $5.90
3. To find how far $8 would go I took 8 and put it into my equation
in the y position, then I algebraicaly solved for x, which was the
amount of miles. my answer was 18 miles
Student 11
1. A linear equation
that fits the given
information is y =
0.35x + 1.70 2. The
cost of a 12-mile ride
would be $5.90. 3. If
you have $8 to spend
on the taxi ride, you
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Extra: First I took $10 and found out how muxh it was without 10%, by
taking 100-10=90. I took .9 or 90% and multifpied it with 10 and
found the answer to be 9. Then I took 9 and put it into my equation
in the y position and solved for x. my answere was about 20.9 miles.
1. linear equation form: y = ax + b
two points used to find equation: (4, 3.10) and (7, 4.15)
let x = the number of miles ridden
let y = the total cost
a = the slope of the equation
Page 7 of 58
can go 18 miles.
the 2 and 1 are sub groups
a = (y2 – y1) / (x2 – x1)
a = (4.15 - 3.10) / (7 – 4)
a = 1.05 / 3
a = 0.35
so far the equation is: y = 0.35x + b
b = the y-intercept of the equation
y = 0.35x + b
substitute point (7, 4.15) for x and y values
4.15 = 0.35(7) + b
4.15 = 2.45 + b
1.70 = b
the equation is: y = 0.35x + 1.70
check by substituting (4, 3.10) into the equation as the values of x
and y
y = 0.35x + 1.70
3.10 = 0.35(4) + 1.70
3.10 = 1.40 + 1.70
3.10 = 3.10
2. substitute 12 as the value of x
y = 0.35x + 1.70
solve for y
y = 0.35(12) + 1.70
y = 4.20 + 1.70
y = 5.90
check by putting 5.90 and 12 into the equation together
5.90 = 0.35(12) + 1.70
5.90 = 4.20 + 1.70
5.90 = 5.90
3. substitute 8 as the value of y
8 = 0.35x + 1.70
6.30 = 0.35x
18 = x
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Page 8 of 58
check by putting 8 and 18 in the equation together
8 = 0.35(18) + 1.70
8 = 6.30 + 1.70
8=8
Student 12
1.) y=.35x+1.7 2.) $
5.90 3.) 18 Miles
Reflection: At first, I did not read the problem carefully and was
slightly confused. Devesh then reminded me that we had reviewed
earlier in the year how to write a proper linear equation. I easily
created two points from the information given, and the equation was
easy to make and solve after that. Sometimes I need to think more
about a problem before I ask for help. This was definitely the
simplest of any POW we have had yet, and I still required help.
Hopefully, on the next POW, I will be able to study and solve the
problem all by myself. This POW is very valid in real life, except
in real life, there is tax and tipping. However, taxi drivers are
just the slightest amount of people using linear equations to make
prices and/or anything else.
1.)Find a linear equation that fits the given information.
(x,y)= (4,3.10)
(x,y)= (7,4.15)
They want us to find a linear equation, if you know two points in a
line you can find the slop.
Our, two points are (4,3.10)
(7,4.15)
1.05 = .35
3.10-4.15 = 4-
7
-3
y=mx=+b
3.10= .35(4) + b
3.10= 1.4 + b
-1.4 -1.4
1.7= b
2.)
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y= .35(x) + 1.7
equation is y= .35(x) + 1.7
you would paid $5.90
Page 9 of 58
y= .35(12) + 1.7
y= 4.2 + 1.7
y= 5.90
3.) 8=.35x + 1.7
-1.7
-1.7
6.3 .35 6.3/.35= 18
.35 .35
8=.35(18) + 1.7
8= 6.3 + 1.7
Extra:
y= m(x) + b
y= .35(x) + 1.7
9= .35(x) + 1.7
-1.7
-1.7
7.3 =
.35x
.35
.35
7.3/.35= 20.85714286
10% of $9 is 90 cents
Student 13
4x+3.10y=x
Student 14
Y=.35X+b
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So you can go 20.85 miles with 9 dollars and still have money to pay
the taxi driver 10% of the cost of the ride.
I really don't understand this problem..I don't see how this can be
a linear equation. I need help with it
Well the first thing that i did to solve this problem is:
X1, Y1 X2, Y2
Page 10 of 58
(4, 3.10) (7,4.15) I set up too find the slope.( Slope= Y2- Y1
Divided by X2- X1)
4.15-3.10=1.05 divided by 7-3=3. And 1.05 divided by 3 is .35
Than the next thing that i need to do it find the "B" and to find
the B i took the first set of parenthasis and pluged them into the
equation, which looked like 3.10=.35(4) i than multiplyed .35 and 4
to get 1.4 than i subtracted 1.4 from 3.10 and got 1.7 which is the
B! so the Equation is now Y=.35X=1.7
Than to find out the answer to part #2 of the question how much a 12mile taxi ride i. I pluged 12 into the X of the equation so it is
now...
Y=.35(12)+1.7
I multiplyed .35 by 12 which = 4.2 and than i added 4.2 plus 1.7
so the cost for a 12 mile taxi cab ride is $5.90
Than to answer part #3 i knew that the total cost was going to be $8
(Y) so i pluged in that to the equation. 8=.35X+1.7
than i subtracted 1.7 from each side...
6.3=.35X
than i divided 6.3 by 3.5 to discover the X.
So X(the total miles that you can drive with $8)
x= 18 miles!
and that is how i discover the answer to this question!
Student 15
1.) the linear equation
that fits the given
information is
y=.35x+1.7 2.) the cost
of a 12-mile taxi ride is
$5.9 3.) if you spend
$8, you can go 18miles bonus.) if you
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1.) y=mx+b (4,3.1)
(7,4.15)
4.15-3.1 = .35 = m 4.5=.35(7)+b
7-4
4.5=2.45+b
-2.45 -2.45
1.7=b
*y=.35x+1.7
2.) y=.35(12)=1.7
*y=$5.9
Page 11 of 58
plan to tip the cab
driver 10% of the cost
of the ride, you can go
about 21-m
Student 16
The linear equation is y
= .35x + 1.7. It would
cost $5.90 to pay for a
12-mile ride. You could
travel 18 miles with
$8.
3.) 8=.35x+1.7
-1.7 -1.7
6.3=.35x
.35 .35
*x=18
bonus.) About 21 miles. If you spend $9 on the ride you pay $9.90
with a 10% tax.
After reading this problem I recorded the key information to ensure
that I understood it thoroughly. I realized that my solution would
have to involve the coordinate plain and graphing since the problem
deals with linear equations.
First I had to define my variables. I knew that x, as the
independant variable, would have to represent the number of miles
travelled, because the amount of miles would affetc the cost. That
leaves y to represent the cost of the ride.
I plotted both points on the coordinate plain just to get a visual.
(4, 3.10) and (7, 4.15) Then I decided to find the slope of these
two. I used the formula
_y2-x2____
y1-x1
The slope turned out to be .35. Now I just needed to find the y
intercept to have a full equation in slope intercept form.Since I
only knew the points and the slope, I realized that I would have to
put my data in point slope form. The point slope formula is
y - y1 = m (x - x1)
I substituted my first point, and slope into the problem so now my
equation looked like this :
y - 3.10 = .35x - 1.4
+ 3.10
+3.10
y= .35x +
1.7
y= .35x + 1.7
Now that I found my linear equation, I can continue on to solve the
other parts of the problem. To find the cost of the 12 mile ride, I
will substitute 12 in for x, which represents distnace.
y= .35(12) +1.7
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Page 12 of 58
y = 4.2 + 1.7
y = 5.9
You would have to pay $5.90 to ride 12 miles
Next to find out how far y ou can travel with $8, I substituted $8 in
for y, which represents the cost.
y = .35x + 1.7
8 = .35x + 1.7
-1.7
6.3 = .35x
18
You could travel 18 miles with $8.
That is how i solved this problem.
Student 17
The linear equation is
y= 0.35x + 1.7 The cost
of a 12 miles ride is $
5.90 I can go as far as
18 miles with $8
A linear equation is represented by
y = ax + b
Where y represents the cost of the ride
Where x represents the number of miles driven
by the taxi
I replace y and x by the numbers given
to find a and b
Equation 1
Equation 2
4.15 = 7a + b
3.10 = 4a + b
By substracting 2 from 1
1.05 = 3a
Therefore
a = 0.35
By replacing the value of a
in Equation1
4.15 = 2.45 + b
Therefore
b = 1.7
The linear Equation is y = 0.35 x + 1.7
The cost of 12 miles ride will be
y = 4.2 + 1.7 = $ 5.9
$ 5.9
With 8$ I will be able to go as far as
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Page 13 of 58
8 = 0.35 x + 1.7
0.35 x = 6.3
x = 18 miles
18 miles
Student 18
1. y=mx+b 2.$5.90 3.18
miles extra. 18.4 miles
REFLECTION:
Wow! this is a pow?? This promblem was fairly simple. I am suprised
they give us a problem this easy after the expanding circle. It was
defenitly a realife to have an easy week. I hope all pow's are this
easy in the future. i had no help from my parents this time becuase
of the simpleness of the problem. I can't wait for spring break. I
hope you have fun!!!
1. y=mx+b
4.15=7m+b - 3.10= 4m+b
1.05=3m divide both sides by 3
m= $.35
3.10=4(.35)+b
3.10=1.40+b
-1.40 -1.40
b=1.70
2. y= .35(12)+1.70
y=$5.90
3. 8=.35(x)+1.70
x=18 miles
Student 19
1. The linear equation
is y=.35x+1.7. 2. The
cost of a 12-mile ride is
$5.90. 3. You can
travel 18 miles if you
have $8. Extra: You can
travel 26.6 miles.
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extra. 10=.35x+.10x+1.70
x=18.4 miles
1. You would use the equation y1-y2= m(x1-x2) to find the answer.
Use 4.15 as your y1 and 3.10 for your y2. Use 7 miles as your x1 and
4 miles as your x2. So, your equation, up to this point, is
4.15-3.10= m(7-4)
You would do your subtraction and get 1.05=3m, then divide both sides
by 3 to get m=.35. Use the equation y=mx+b to get the linear
equation, y=.35x+1.7
Page 14 of 58
2. To find the cost of a 12-mile ride, you would plug in 12 to the
equation. y=.35(12)+1.7. So, y=$5.90, which is the cost of a 12-mile
ride.
3. You would use the equation, y1-y2=m(x1-x2). Again, you would plug
in the numbers from the problem to get 4.15-y2=.35(7-0). Next, you
would multiply the equation out and get 4.15-y2=2.45. Then, you would
do -y2=-1.7, and when you divide that out you get y=1.7. y=the cost,
so 8=.35x+1.7 then subtract 8 on both sides of the equation and you
get .35x=6.3. Divide both sides of the equation by .35 and you get
x=18 miles.
Student 20
1. y=.35x+1.70 2. $5.90
3. 18 miles ec. 21.12
miles
EXTRA: First, 10%of 10 is 1, so 10(dollars you have)+1=11. Put 11
into the equation and get 11=.35x+1.7 and subtract both sides of the
equation by 11 and you get 9.3=.35x. Next, you divide both sides of
the equation by .35 and you get x= 26.6 miles.
1. 4.15-3.10/7-3=.35
3.10=.35*4+b
3.10=1.4+b
3.10-1.4=b
1.7=b
y=.35x+1.7
2. y=.35*12+1.7
y=5.9
$5.90
3. 8=.35x+1.7
8-1.7=.35x
6.3=.35x
6.3/.35=x
18=x
18 miles on $8
Student 21
An equation that fits
the given information
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ec. y=(.35x+1.7)1.1
y=21.12
21.12 miles
Solution for question:
Page 15 of 58
is y= .35x + 1.7 . The
cost of a twelve mile
ride is $5.90 . You can
go 18 miles if you had
$8 dollars to spend on
the ride. For the extra,
you would be able to
go 20.9 miles.
1. The first thing that I did to solve this equation is I found the
coordinates of (3.10,4) and (4.15,7). Then I needed to find the
slope of the line since it is a linear equation so I subtracted
them. So, so far I had y= .35x + b. I needed to find the y-intercept
of this equation, so I substituted one of the points into the
equation and got y= .35x + 1.7 .
2. The first thing I did to solve this question was I inputed the
value of 12 into the equation because it asks how much would it be
for a 12 mile ride and I came up with $5.90.
3. I did the same thing with this question as I did with the last
question only the number 8 that I substituted went into the y place.
I recieved an answer of 18 miles.
Student 22
1.) The linear equation
that fits the given
information is x + (m)
(y). 2.) The cost for a
12-mile ride is $5.90.
3.) You can go 18 miles
if you have $8 to
spend. Extra: You can
go about 20.9 miles
with $10 and with a
10% tip.
Extra Credit. The first thing that I did for this equation is I
found out that ten percent of ten dollars is one dollar and
subracted that from my taxi budget because I have to tip the driver
10 percent. Then I just put nine dollars into my equation and popped
out with 20.9 miles.
The Problem
The first thing that we did was read the problem and understand what
we had to do next. We also looked over the given information about
Lineville.
Problem #1
To write a linear equation that fits the given information, fist we
had to define our variables. We prefer to use x and y.
x = the initial cost
y = the rate per mile
We knew that there had to be a rate per mile because when we divided
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the cost by the mile of the 4-mile ride and for the 7-mile ride, they
came out to be different. That means that there must be a rate you
must pay even if you go less than one mile. Now we had to put the
information for the 4 and 7 mile ride into equations.
x + 4y = 3.1
x + 7y = 4.15
To figure out what the cost of x and y are we knew that
we had to do linear combination. But we would have to cancel
something out to find one variable first. We decided to cancel out
the x because it is just easier to cancel than the y.
To do this we had to multiply one of the equations through by –1. We
wanted to use the first equation that we had written because
everything would stay positive and we prefer positive.
-1 (x + 4y = 3.1)
-x – 4y = -3.1
Now we where able to use linear combination to combine the two
problems.
-x – 4y = -3.1
+
x + 7y = 4.15
= 3y = 1.05
Then we just simplified the rest by dividing both sides by 3 to find
the value of y.
3/(3y) = (1.05)/3
y = 0.35
So now we figured out that you have to pay 35 cents per mile. To now
find the initial rate we had to substitute that information into one
of the original equations and simplify it. You can use either one of
the fist equations to find the initial cost.
x + 4 (0.35) = 3.1
x + 1.4 = 3.1
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-1.4 -1.4
x = 1.7
So the initial cost for riding the taxi is $1.70, even if it’s less
than a mile. Now that we had the cost for every mile, and the initial
cost, we can solve the next problem easily.
Problem #2
Finding the cost of a 12-mile ride was simple once we knew what the
initial rate and the rate per mile is. We used the same equation that
we had set up in problem #1, the only difference is that we filled in
the variables and needed to simplify to find the cost.
1.70 + 12(0.35) = 3.1
1.70 + 4.2 =
= 5.9
So after simplifying by using the distributive property we found that
it costs $5.90 for a 12-mile ride.
Problem #2
For this problem we still needed to use the same information but we
realized that we needed a new variable. We had to find how far you
could go for $8 so the new variable that we needed was for the number
of miles.
m = the number of miles driven
So the equation that we would use now is…
x + (m)(y) = the total cost
(This is when we realized that this was the answer to the whole first
problem and what we did was just show how it worked with the
examples.) So we had to plug in the appropriate information for this
problem and solve for m.
1.70 + m(0.35) = 8.00
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-1.70
-1.70
m(0.35) = 6.3
0.35
m = 18
After simplifying using the distributive property, we found that you
can travel 18 miles for $8.00.
Extra:
This problem was very much like the last problem in terms of
simplifying but there was one extra step, which was the tip for the
cab driver. But the only thing that we had to do differently was find
the tip for the money and subtract it from $10.00. We knew that 10%
of $10 was one and we also knew that 10 – 1 = 9 so we had to
figure out how far someone could go with $9. We set up the equation
and simplified it.
1.70 + m(0.35) = 9.00
-1.70
-1.70
m(0.35) = 7.3
0.35
m = 20.857142….(Rounded to the nearest tenth is 20.9)
So you can go about 20.9 miles if you had $10 and wanted to give the
driver a 10% tip.
Reflection
At fist we honestly had a little difficulty with this problem. We did
not know exactly what to do right away even though we have practiced
with linear equations and linear combination in algebra. (And that
was last chapter!) But when we finally figured out what to do, it was
pretty strait forward.
We did not realize that we had done the whole first portion right
until we started typing this! We realized that we did not in fact
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answer the question correctly but instead we used the linear equation
in action with the examples. We kept what we typed though because
what we did also helped us with problem #2 and problem #3.
The skills that we needed to figure out this weeks PoW we had already
learned a wile ago. We needed knowledge of linear equations and
linear combination. The only other thing that we really needed was
knowing the distributive property! After doing these problem we
checked and rechecked our calculations and our problem solving
strategies and we think that our solutions make sense. We could not
think of any other ways to do the problem except maybe use the other
equation towards the end in Problem #1 (which we did afterward and
still came out with the same solutions)
Ashford School
Group #7
Student 23
Student 24
1.) y=.35x+$1.70 2.)
The cost of a 12 mile
ride would be $5.90 3.)
You can go 18 miles if
you have $8 to spend
on a taxi ride. Extra)
You can go about 21.1
miles if you had $10 to
spend and plan to tip
the cab driver 10% of
the cost of the rid
1. The linear equation
is: y = 0.35x + 1.7 with
y is the taxi cost, x is
the distance 2. The
cost of 12-mile ride is
$5.9 3. If I have $8 to
spend on the taxi ride,
I can go for 18 mile.
I used the points (4,3.10) ; (7,4.15)
I subtracted y2 from y1 and x2 from x1 to get 1.05 divided by 3
which equals .35
I then used the equation y=mx+b to find my answers for problems 1,
2, and 3
First, the the taxi rates are based on a linear equation, so I assume
that the formula of the taxi rate is the same as the formula of
linear equation.
y = ax + b
I submit the information given above. Let the taxi cost be y and
the distance be x.
(i) cost for a 4-mile ride is $3.10 --> y = 3.1 x = 4
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3.1 = 4a + b
(ii) cost for a 7-mile ride is $4.15 --> y = 4.15 x = 7
4.15 = 7a + b
As a result, I have 2 mathematical equations.
The first is 3.1 = 4a + b and the second is 4.15 = 7a + b.
Then, I eliminate the first equation with the second equation to get
the value of a.
(i) ... 3.1 = 4a + b
(ii)... 4.15 = 7a + b (-)
-1.05 = -3a
a = 0.35
Then I substitute the value of a to the first mathematical equation
to get the value of b.
3.1 = 4a + b
a=0.35 -> 3.1 = 4(0.35) + b
3.1 = 1.4 + b
b = 1.7
Thus, the linear equation is: y = 0.35x +1.7 with y is the taxi
cost and x is the distance.
Second, to find the cost of a 12-mile ride, I substitute the distance
as the value of x to the equation from the answer of the first
question to find the cost as value of y.
y = 0.35x + 1.7
x = 12 -> y = 0.35(12) + 1.7
y = 4.2 + 1.7
y = 5.9
Thus, the taxi cost of a 12-mile ride is $5.9
Third, to find how far can I go if I have $8 to spend on the taxi
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ride, I substitute the cost as value of y to the equation from the
aswer of the first question to find the distance as value of x.
y = 0.35x + 1.7
y = 8 -> 8 = 0.35x + 1.7
0.35x = 6.3
x = 18
Thus, the distance I can go with $8 is 18 mile
Extra:
I plan to tip the cab driver 10% of the cost of the ride, so the cost
to expand will be 110 percent from the original cost. To find the
distance, I substitude the cost as value of y to the
equation from the answer of the first question.
Let start. The equation is:
y = 0.35x + 1.7
I plan to tip the driver, so the equation will be:
y = 110% (0.35x + 1.7)
Total money I spend for the driver and the taxi cost is $10 -> y = 10
10 = 110% (0.35x + 1.7)
100 = 11 (0.35x + 1.7)
100 = 3.85x + 18.7
3.85x = 81.3
x = 21.12
Student 25
1. The equation used
for the information is:
y = .35x + 1.7 2. The
cost of a 12-mile ride
would be $5.90. 3.
With $8, you could
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Thus, I can go for 21.1 mile.
EXPLANATION
___________
Now that I am approaching a new problem in my POW, I face a unique
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travel 18 miles.
challenge consisting of multiple steps.
1. At my first attempt to solve this equation, I used the basic
equation of:
y = mx + b
y = total cost
m = cost per mile
x = distance in miles
b = initial cost for taxi
After concluding to a body equation, I plugged in the information I
knew into the equation:
3.10 = 4m + b
4.15 = 7m + b
My next step in solving for an equation was to combine the two
equations I had. To do this I would subtract one from the other:
4.15 = 7m + b
- 3.10 = 4m + b
------------1.05 = 3m
m = 1.05/3
m = .35
Once I knew the cost per mile, I would need to find the initial cost
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of the taxi ride. To do this I would simply substitute the value for
(m) into one of the equations.
3.10 = .35(4) + b
3.10 = 1.4 + b
b = 3.10/1.4
b = 1.7
To check this I would substitute my values into the other equation.
4.15 = .35(7) + 1.7
4.15 = 2.45 + 1.7
4.15 = 4.15
Now that I knew these were the correct values for the variables, I
knew that my final equation would be:
y = .35x + 1.7
2. To solve this problem, I would simply just substitute 12 miles in
for the variable x.
y = .35x + 1.7
y = .35(12) + 1.7
y = 4.2 + 1.7
y = 5.9
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y = $5.90
3. In this problem, I would substitute in $8 in for the variable y.
y = .35x + 1.7
8 = .35x + 1.7
6.3 = .35x
18 = x
x = 18 miles
EXTRA
_____
In order to solve this problem, it would be necessary to factor in
the tip along with the equation. To do this, I would subtract the
tip and find how far I could go with the remaining money:
.1(10) = 1
10 - 1 = 9
The money that would be remaining after the tip would be $9. From
here I would solve to find how far I could ride with that.
y = .35x + 1.7
9 = .35x + 1.7
7.3 = .35x
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20.9 = x
You could travel 20.9 miles with ten dollars including a tip.
Student 26
Explained in the
explanation
1. the linear equation is y = 2.86x - 4.9
2. $4.38
3. 25.98 mi
This is a fairly simple problem.I used y = mx + b
first I had to review my math book to make sure I knew what each
variable was. I confirmed that, x and y were the points, m is the
slope and b is the y intercept.
The point is the distance and cost.
For the m you must subtract the y2 and y1 from each other and divide
that by x2 - x1
I got 2.86
So i used a point for to get b
4 = 2.86(3.10) + b
I just figured out what b had to be by figuring out the rest of the
equation.
I got b to be -4.9
Student 27
1. y = .35x + 1.7 2.
$5.90 3. 18 miles
That is how I solved the equation.
1. During 3 miles, the cost increased by $1.05
4.15 - 3.10 = 1.05
So if I divide 1.05 by 3, I can find out how much it costs per mile.
1.05 / 3 = .35
So it's 35 cents per hour.
.35(4) = 1.40
It costs $1.40 to travel 4 miles. But it said $3.10, so
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3.10 - 1.40 = 1.70
$1.70 is the starting price.
So the equation for the taxi is,
let x represent the number of miles,
y = .35x + 1.70
I checked the equation with 7-mile ride this time.
.35(7) + 1.70
2.45 + 1.70 = 4.15
I got $4.15 which is the right answer.
So the answer for question number 1 is,
y = .35x + 1.7
2. I substituted 12 in to x.
y = .35x + 1.7
y = .35(12) + 1.7
y = 4.2 + 1.7 = 5.9
Total price is $5.90.
3. I substituted 8 into y, because it was the total price.
8 = .35x + 1.7
6.3 = .35x
6.3 / .35 = x
x = 18
You can go 18 miles with 8 dollars.
I checked if my answer is right.
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.35(18) + 1.7
6.3 + 1.7
=8
Reflection:
First, I just divided 3.10 by 4 and 4.15 by 7 without thinking about
the basic price. Obviously, it didn't work out so I asked my dad to
help me. He went throught the equations and I understood. I think
the question was tricky, because it looks easy but actually
confusing.
Student 28
1. My linear equation
is y=0.35x+1.70 2. A
12-mile ride would
cost $5.90 3. With $8, I
could go 18 miles.
To solve my answers for these problems, I first tried writing an
equation to solve both values given. It asked for a linear equation
and that is y=mx+b, so I knew there was a flat rate and a price per
mile.
3.1=4x+b
4.15=7x+b
4.15-3.1=1.05
I know had a way to solve both equations simultaneously:
4x+b+1.05=7x+b
I subtracted each side from b, and got:
4x+1.05=7x
I know subtracted each side by 4x and got:
1.05=3x
I then divided by 3
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0.35=x
I know had solved for x, or the rate for one hour. The rate is 35
cents an hour, but know I had to find the flat rate, so I plugged
and checked:
3.10=4(.35)+b
3.10=1.4+b
1.7=b
I know tried this with the other equation to see if it was similar.
4.15=7(.35)+b
4.15+2.45+b
1.7=b
This was a way of checking my answer and the flat rate of the taxi
drive distance. Now that I had my linear equation set up,
y=.35x+1.7 with x being the miles, I could know solve the cost of a
12 mile ride:
y=.35(12)+1.7
y=4.2+1.7
y=5.9
For a 12 mile drive in this taxi, it would cost $5.90.
I know had to try and see how many miles I could go with $8. For
this I knew y, but needed to find x. I set up my equation like this:
8=0.35x+1.7
6.3=0.35x
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18=x
I could go 18 miles with $8.
Now that I had all I needed to know figured out, I went back and
checked all my work again. This was probably one of the easiest
POW's for me because I knew how to start right away. I think the
work we did with linear equations helped a lot in solving this
problem.
Student 29
1.) The linear equation
is y = 0.35x +1.7. 2.)
The cost of a 12 mile
ride is $5.90. 3.) You
can go 18 miles when
you have $8.00 to
spend.
P.S. Mr. Compton, did you get my last POW that I emailed to you
last Monday, I haven’t seen it in my CAF. Thanks!
Here is how I worked out the three problems to get my answers:
--------------------------------------------------------------Problem #1
x = miles
y = dollars
x| 4 | 7 |
----------------y | 3.10 | 4.15|
(numbers are subunits)
|
v
y1 - y2 (4.15 - 3.10) 1.05
--------- = ------------- = ------ = 0.35
x1 - x2
(7 - 4)
3
y = mx+ b
m = 0.35
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3.1 - 0.35 - 0.35 - 0.35 - 0.35 = 1.7
y = mx + b
b = 1.7
The equation is y = 0.35x + 1.7.
----------------------------------------------------------Problem #2
I had to use my equation to see how much it would cost for a 12 mile
ride.
y = 0.35(12) + 1.7
y = 4.2 + 1.7
y = $5.90
It would cost $5.90 for a 12 mile ride.
----------------------------------------------------------Problem #3
In this problem, the setup is somewhat reversed compared to the
problem #2 setup.
y=8
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8 = 0.35(x) + 1.7
6.3 = 0.35x
18 = x
You could go 18 miles when you have $8.00 to spend.
--------------------------------------------------------Extra:
Student 30
1. The equation is
Y=0.35X+1.7 2.
Y=0.35(12)+1.7
Y=$5.90 3. Y=$8.00
$8.00=0.35(X) +1.7
X=18 miles
You could go approximately 21.1 miles.
4 miles= $3.10
7 miles= $4.15
1. X=miles
Y=dollars
A linear equation is Y=mX+b by definition so from there I built the
basis of my problem solving
X
Y
4
3.10
7
4.15
Y1-Y2/X1-X2=(4.15-3.10)/(7-4)=0.35
m=0.35
b=0 term
3.10-0.35=2.75
2.75-0.35=2.4
2.4-0.35=2.05
2.05-0.35=1.7
The equation is Y=0.35X+1.7
2. Y=0.35(12)+1.7
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Y=$5.90
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3. Y=$8.00
$8.00=0.35(X) +1.7
X=18 miles
I enjoyed this problem because it was straight and to the point. I
thoroughly enjoyed that this problem was a real life problem and
that it used algebra in a realistic situation. I could see a person
doing something like this in their daily life as a corporate worker
and that made me eager to participate and solve the riddle of this
problem. I had some trouble on the extra but my sister showed me
how to start off. I was having trouble with the 10% and then my
sister showed me that my spending money would end up being $9.00.
From there it was a piece of cake. All and all I really enjoyed
this problem and would be thrilled to see something similar in the
future.
Student 31
1) my linear equation
is C= .35D + 1.7 2) a 12
mile cab ride is $5.90
3)with 8 dollars you
can go 18 miles.
Student 32
1) The linear equation
that fits the given
information is y =
0.35x + 1.7 2) The cost
of a 12-mile ride is
$5.90 3) You can go 18
miles if you have $8 to
spend on the taxi ride.
Extra: 21.1 miles
first i put it into a form of a linear equation. using c as cost and
d as distance. then for the 2nd answer i plugged 12 miles in for
distance and then solved accordingly to get $5.90 which is how much a
12 dollar cab ride is. then for the third question i plugged 8
dollars in for cost and solved accordingly to get 18 miles. that
means that it takes 18 miles to pay 8 dollars in the cab.
#1:
First, I found the cost for a one-mile ride. I did this by using the
information given to make two points (as if they were on a graph) and
find the slope of the line that would connect the points. I imagined
that the x values in the ordered pairs would be the length of the
ride, and that the y values in the ordered pairs would be the cost of
the ride. So, because a four-mile ride costs $3.10, the point
representing that information is (4, 3.1). A seven-mile ride costs
$4.15 and is represented by the ordered pair (7, 4.15).
Next, I found the slop of the line that would connect the two points.
The formula for finding slope is:
(y2 - y1) / (x2 - x1)
Then I substituted in the numbers from the two ordered pairs and
solved.
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(4.15 - 3.1) / (7 - 4) = 1.05 / 3 = 0.35
So, each mile costs $0.35. After that, I decided to check my answer
to see if it worked. I took this price per mile and multiplied it by
the two lengths of the given rides.
0.35 * 4 = 1.4
1.4 does not equal 3.1
0.35 * 7 = 2.45
2.45 does not equal 4.15
This showed that there must be another part to the linear equation
other than the price per mile. Then I remembered that sometimes taxi
drivers charge a set fee. I also remembered the equation for linear
lines.
y = ax + b
In this equation b is always constant, just like the set price that
taxis charge.
Let y = the cost of the taxi ride
a = the price per mile
x = the number of miles driven
b = the set price
Then, I substituted in values for y, a, and x and solved for b.
y = ax + b
3.1 = 0.35*4 + b
3.1 = 1.4 + b
b = 1.7
So, the set price that the taxis charge is $1.70.
The linear equation is y = 0.35x + 1.7
#2
After finding the linear equation, I simply substituted in 12 miles
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for x and solved for y to find the cost of a twelve-mile ride.
y = 0.35x + 1.7
y = 0.35*12 + 1.7
y = 4.2 + 1.7
y = 5.9
The cost for a twelve-mile ride is $5.90.
#3
For this problem, I did just the opposite of number two. I
substituted in $8 for y and solved for x to find how far I could go.
y = 0.35x + 1.7
8 = 0.35x + 1.7
6.3 = 0.35x
18 = x
I could go 18 miles with $8.
Reflection:
This was definitely one of the easiest problems for me. I was quick
to realize how to solve it by making two ordered pairs, finding the
slope, and using the linear equation y = ax + b. For people who
didn't realize that right away, however, it would have been harder to
solve. The extra was more difficult, and it took me quite a few
trials before I figured out how to do it. At first I tried a couple
different methods and checked my answers, but none of them seemed to
work. But after a while, I was able to discover how to go about
solving it. The twist was that you have $10, but you have to tip the
driver 10% of the cost of the ride. So, you couldn't spend $10 on the
ride.
Extra:
To solve this, I set up an equation to find the amount of money I
could spend with a tip of 10%.
0.1y + y = 10
y + 10y = 100
11y = 100
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y = 9.1
Then, I substituted the price of the ride into the original equation
and solved for x.
y = 0.35x + 1.7
9.1 = 0.35x + 1.7
7.4 = 0.35x
21.1 = x
You can go about 21.1 miles if you have $10 to spend and tip the
driver 10% of the cost.
Student 33
1. Y = $.35x + $1.70 2.
A 12-mile cab ride
would cost $5.90. 3.
For $8, you can go on
an 18-mile cab ride.
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First I realized the linear equation would need to be Y = MX
+ B. M is the y-intercept and B is the slope. So then I figured out
the M very easily because I subtracted 4 miles from 7 miles to get 3.
Then I divided that by $1.05 because $4.15 minus $3.10 equals $1.05.
Thus $1.05 divided by 3 equals $.35. I then knew that the yintercept was $.35 or M. So I know that Y = $0.35x + B.
Then to figure out B, I used the equation y1- y2 = m (x1-x2). This
is because I have many variables. So I decided to plug in the data
for the 4-mile ride costing $3.10. I used y2=$3.10 and x2=4miles:
y-$3.10= $.35(x-4) (distribute the $.35)
y-$3.10 = $.35x - $1.40
To get Y by itself on one side of the equation, I added $3.10 from
each side:
y = $.35x - $1.40 + $3.10
y =$ .35x +$1.70
So therefore the answer to the first question was Y = $.35x + $1.70
For the next problem, I figured out that it is $0.35 for an
extra mile after the 4 miles. I subtract the 4 miles from the 12
miles because the first 4 mile ride has a fixed value. I am left with
8 so I multiply $0.35 by 8miles to get $2.80. I then need to add
back in the $3.10 for the cost of the first 4 mile ride. This means
it would cost $3.10 + $2.80 or $ 5.90 to go 12 miles.
For the third problem, I subtracted $3.10 from the $8.00
because that is the fixed price for the first 4 miles. That equals
$4.90. Since it is $0.35 for each additional mile, I divided $4.90 by
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Student 34
The linear equation
that fits this problem is
y = 0.35x+1.70. A 12
mile ride costs $5.90,
and you can go 18
miles, if you spend
$8.00 on a taxi ride. If
you have $10.00 to
spend on a ride and
you want to tip the
driver, you can travel
about 21 mile
$0.35, which equals 14. So for $8 you can get an 18- mile cab ride.
Reflection:
Since I had already learned and reviewed this material, it came
easier for me than some of the other problems. I am glad I knew the
formulas or it might have been harder for me. I like word problems so
I enjoyed doing this problem. It was fun to do and I hope to get more
similar to this one!
To find a linear equation, you must know the slope and y intercept.
To find the slope, you must divide the difference of the "y’s", which
are the costs of the 4 and 7 mile rides, by the "x’s", which are the
distances, 4 and 7. When you subtract the y’s, $4.15, the cost of the
7 mile ride, by $3.10, the cost of the 4 mile ride, you get $1.05.
When you subtract the "x’s", 7, the distance you go when you pay
$4.15, by 4, the distance you can go for $3.15, you get 3. When you
divide $1.05, the difference of the "y’"s by 3, the difference of
the "x’s", you get $0.35, the slope.
$4.15 - $3.10 = $1.05
7-4 = 3
$1.05/3 = 0.35
Then you plug $0.35, the slope, into “y = mx+b”, the format of a
linear equation. “M” represents the slope, so you insert 0.35 in
the “m” spot. Then I plugged one of the costs of rides in the “y”
part, and also, I plugged one of the distances in for “x”. I did this
so I could solve for “b”, the “y” intercept. When I solved, I got
that “b” equals 1.70. This means that the equation is y, the amount
of money spent, equals 0.35, the slope, times x, the distance
traveled, plus 1.70, the added price for getting in a taxi.
$4.15 = 0.35 (7) + b
$4.15 = 2.45 + b
-2.45 -2.45
-------------------1.70 = b
Final Equation: y = 0.35x + 1.70
To find the cost of a 12 mile ride, I substituted 12 for “x” in the
equation, since “x” represents the distance. When I solved the
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equation, I got $5.90, which is the cost of a 12 mile ride.
y = 0.35(12) + 1.70
y = 4.20 + 1.70
y = 5.90
To find how far you can go with $8.00, you plug 8 in for “y”,
since “y” is the amount of money the ride costs, and then solve
for “x”, the distance. When I finished solving, I got that “x” is 18
miles. You can go 18 miles with $8.00.
8.00 = 0.35x =1.70
-1.70
-1.70
------------------------------6.30 = 0.35x
----- -------0.35
0.35
18 = x
If you drive “x” miles, the cost is $0.35 cents a mile, plus $1.70.
However, if you have a 10 percent tip, you have to multiply the cost
of the trip, $0.35x +$1.70, by 1.1, to add for the 10 percent tip.
You have to solve the equation, 1.1(0.35x + 1.70) = $10.00, to get
the amount of miles you can go, if you spend $10.00 and tip the
driver 10 percent. Then you solve for “x”, to get the amount of miles
you can ride. You get about 21.1 miles. The answer is that you can
travel about 21.1 miles, if you have $10.00 to spend and you tip the
driver 10 percent of the cost of the ride.
Student 35
Twelve miles equals
$4.76. With $8.00, you
can go 36 miles.
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1.1(0.35x + 1.70) = 10.00
0.385x + 1.87 = 10.00
- 1.87 -1.87
0.385x = 8.13
------ -----0.385 0.385
x = 21.11688312, which rounds to 21.1 to the nearest tenth.
First we figured that the first five miles were more expensive than
additional miles. Each mile up to $5.00 costs $ .77. Each
additional mile costs $ .13. So in order to figure out how much 12
Page 38 of 58
Student 36
1.) y=.35x+1.7 2.)
y=5.90 3.)18miles=x
extra 20.9 miles
miles cost, we mulitiplied 5 times .77 and 7 times .13 and got $4.76.
To figure out how far we could go with$8.00 we determined that the
first five miles were $3.85. Then we subtracted 8.00 from $3.85. That
was $4.15. Next we divided $4.15 by 13. That was 31miles. Last, we
added 5 miles to the 31 miles and that was 36 miles so you can go 36
miles with $8.00.
1.) 4.15-3.1 equals 1.05 equals 3 miles. take 1.05 divided by 3 and
get .35 cents per mile.
4.15-(.35times7)=1.7 dollars
2.) y=.35(12) =1.7
y equals .35x+1.7
y=4.2+1.7
y=5.90 dollars
3.) 8=.35times1.7
subtarct 1.7 from each side and get 6.3 equals .35
then divide 6.3 equals .35x
the answer is 18 miles=x
Student 37
1. y= 0.35x+1.7 2. It
cost $5.90 for 12 miles
Il coût 5.90$ pour 12
milles 3. We can make
18 miles with $8 Avec
8$ on peut faire 8
milles en taxi extra:
We can make 21.12
miles if we $10 and we
give 10% of tips Si l'on
a 10$ et on donn
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extra 10;9=9
9 equaks .35xplus 1.7
subtract 1.7 from each side and then divude 7.3 which eequals .25x
divide 7.3 from eachside and get 20.9 miles!
1. y1-y2 3.10-4.15 -1.05
----- = ---------- = ----- = 0.35
x1-x2 4 - 7 -3
y=0.35x+b
3.10=0.35(4)+b
b=1.7
y=0.35x+1.7
2. x=12
y=0.35(12)+1.7
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y=5.90
3. y=8
8=0.35x+1.7
-1.7 -1.7
6.3 = 0.35x
---- ----0.35 0.35
18=x
Student 38
The equation for the 4
mie ride is; 3.10 = 4x +
b, and the equation for
the 7 mile ride is 4.15 =
7x + b. The cost for a
12 mile ride would be
$5.90, and with 8
dollars you can go 18
miles.
Extra: y = 10-0.1y
+0.1y +0.1y
1.1y = 10
---- --1.1 1.1
y = 100/11
100/11 = 0.35x+1.7
-1.7
-1.7
815/110=0.35x
------- ----0.35 0.35
1626/77 = x
21.11688312 = x
This Problem of the Week, Taxi!, is asking us to solve a real-world
problem with a linear equation. This is very practical, because there
are many situations in which this is helpful.
The first thing I tried, was dividing $3.10 by 4 miles, and $4.15 by
7 miles, but it does not work. You can double check your work, by
finding the rate per mile and multiplying it by the number of miles
driver. Alas, you cannot forget about a 'base' rate, or a fee that
starts for just getting into the taxi. You also need to know what a
Linear Equation is. A Linear Equation, is written in the form:
y = mx + b
We need to define our variables before we start. y is the total cost,
m is the number of miles driven, and x is the rate per mile. b is
the 'base' cost. We can substitute for the known variables.
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Page 40 of 58
"The cost for a 4-mile ride is $3.10" this in an equation form is:
3.10 = 4x + b
"The cost for a 7-mile ride is $4.15" this in an equation form is
4.15 = 7x + b
To solve this equations I took the following steps;
y = mx + b
3.10 = 4x + b Substitute
3.10 - 4x = b Subtract '4x' from both sides.
We have solved for b, and we can substitute it in for our second
equation.
y = mx + b
4.15 = 7x + b
4.15 - 7x = b Subtract '7x' from both sides.
4.15 - 7x = 3.10 - 4x Substitute for b.
4.15 - 3.10 = 7x - 4x Commute.
1.05 = 3x
Arithmetic.
x = 0.35
We now know that the rate per mile is 35 cents. However we need to go
back to the first equation to solve for b, or our 'base' rate.
3.10 = 4 * 0.35 + b
3.10 = 1.40 + b
1.70 = b.
We have successfully solved for x and b. We can move on to the second
part of the equation.
y = mx + b
y = 12 * 0.35 + 1.70
y = 4.20 + 1.70
y = 5.90
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Page 41 of 58
The cost for a 12-mile ride would be 5.90 cents.
Now we have to reverse the process we completed in step 2, because we
know the price, but not the distance.
y = mx + b
8.00 = m * 0.35 + 1.70
6.30 = m * 0.35
Subtract 1.70 from both sides.
6.30 / 0.35 = m
m = 18
With 8 dollars, you can ride 18 miles.
Reflection:
This problem, unlike the recent problems focused on material that was
learnt along time ago. I enjoyed working it because I know of the
many practical applications, such as gas, or labor, and I knew
exactly how to do it. Additionally, it is something that I have a
very good chance of using it, with whichever job I have. Greeat
Problem!
Student 39
The linear equation is:
y = 0.35x + 1.7 where y
= the cost of the taxi
trip, and x = miles
traveled. The cost of a
12 mile ride is $5.90.
With $8, you can travel
18 miles. The extra
credit answer is about
21.1 miles.
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When I first saw this problem I wrote down everything to make it
easier to solve. I wrote down "4 miles: $3.10, 7 miles: $4.15".
After this I used guess and check to find my equation. The way I
did this is I took the value 3.10, from the price of the 4 mile
trip, and I subtracted from my guess of the initial fee. When I
subtracted that I took the remaining and divided that by 4, getting
the cost per mile. I multiplied that cost by 7, for the other trip,
and added my initial fee, which, in my hopes got me 4.15.
I finally got the correct equation by guessing and the
equation was y = .35x + 1.7. I checked this with both equations.
I used this equation to find the cost of the 12 mile ride,
by substituting the x variable in the equation. It looked like
this: y = .35x + 1.7
y = .35(12) + 1.7
y = 4.2 +1.7
y = 5.9
Page 42 of 58
cost = $5.90
For the next question I just substituted, but put the final cost in
the y position.
y = .35x + 1.7
8 = .35x + 1.7
6.3 = .35x
18 = x
miles you can go on $8 = 18
Student 40
The linear equation is
Y = 1. 0 5 X + 1 . 7
Useing my equation to
find the cost is $ 5 . 9 If
I had only $ 8 i can go
18 miles
This question was not too difficult for me, once I guessed
the equation. I didn't know how to find it correctly using Algebra
so I just guessed. This problem was easy for me overall though,
because we worked on these linear equations and substituting problem
alot during the last few chapters.
Y=$
X = Miles
M = slope
B = Y intersept mi = Miles
To find the linear equation for Y = M X + B
I have two points ( 4 , 3.10 ) and ( 7 , 4. 1 5 )
useing Y1 - Y 2 / X 1 - X 2 i could find the slope
$ 4 . 1 5 - $ 3 . 1 0 / 7 mi - 4 mi = $ 1 . 0 5 / 3 mi = M
To find B plug in a pint
$ 3 . 1 0 = $ 1 . 05 / 3 mi ( 4 ) + B
$ 3 . 10 = $ 1 . 4 / mi + B
$ 1. 70 / mi = B
so my equation is Y = $1 . 0 5 / 3 X mi + $ 1 . 70
useing the equation to find the cost of a 12 Miles ride:
Y = ( $ 1 . 0 5 / 3 mi ) ( 1 2 mi ) + $ 1 . 70
Y = $ 4 . 20 + $ 1 . 70
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Page 43 of 58
Y = $ 5 . 90
If only $ 8 to spend for a taxi ride so i just plug $ 8 in the equation:
$ 8 =( $ 1 . 0 5 / 3 mi ) ( X mi ) + $ 1 . 7
$ 6 . 3 = ( $1 . 0 5 / 3 mi ) ( X mi )
$ 1 8 . 9 mi = $ 1 . 0 5 X mi
18 mi = X
This problem was easy! my dad didn't have to help me at all! just a bit
the only problem was i got a bit mixed up with which one was the dollor sign
and which one was the miles sign but my dad helped me straiten that out and
thats it.
this problem not so boring like the othere problems this one was sorda
intresting well not as fun as watching TV or playing with you're friend but a
intresting math problem.
Student 41
1. 0.35x + 1.7 2. $4.20
3. 18 miles
This was really easy compared to the before POWs. I could do this
really fast because I spent lot of times using linear equation in
math class. I think if you know the equation of the linear equations,
it shouldn't take you a long time to solve.
1. I used: y = mx + b
Then, I plugged in the given information to the equation.
3.10 = m4 + b
4.15 = m7 + b
Next, I solved for b:
b = 3.10 - m4
b = 4.15 - m7
In order to get rid of b, I combined them.
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3.10 - m4 = 4.15 - m7
-m4 + m7 = 4.15 - 3.10
m3 = 1.05
m = 0.35
After finding m, I plugged in 0.36 to m and solved for b to check
0.35 * 7 + 1.7 = 4.15
b = 1.7
Final equation is : y = 0.35x + 1.7
2. Using the equation from number 1, I just plugged 12 into x
y = 0.35*12 + 17
y = 4.20 + 1.7
y = 5.70
The cost of 12-mile ride is $ 5.70
3. I also used the equation from number 1, but this time, I plugged 8
into y.
y = 0.35x + 1.7
8 = 0.35x + 1.7
6.3 = 0.35x
18 = x
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You can go 18 miles if you have $8 to spend on the taxi ride.
I realized that you can't forget something you've learned from the
past in order to go onto the next step, for example, like linear
equation.
Student 42
My equation is y =
0.35x + 1.7, it takes
$5.90 for 12 miles, and
you can go 18 miles for
$8.
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In order to figure out an equation, I had to find out how much money
was charged per mile. Instead of dividing $3.10 by 4, I decided to
subtract $3.10 from $4.15 because I knew there would probably be a
minimum fee.
$4.15 - $3.10 = $1.05 for 3 miles
So now I can divide 1.05 by 3 to get the amount of money charged per
mile. I get that $0.35 is charged per mile.
Now I have to find out the amount of the minimum charge so I use the
exaple of $3.10 for 4 miles. I already know that it's $0.35 per
mile, so I mulitply that by 4 (for the amount of miles) and subtract
from $3.10 to get the minimum fee of $1.70.
Now I can put together my linear equation of y = 0.35x + 1.7.
(y=amount of money charged and x=amount of miles)
I have to find out how much money is charged for a 12 mile ride, so
I substitue 12 for x and get
y = 0.35 (12) + 1.7
y = 4.2 + 1.7
y = 5.9
Therefore the cost of riding for 12 miles is $5.90.
Now in order to find out how many miles I can ride for $8, I simply
put 8 in for y.
8 = 0.35x + 1.7
6.3 = 0.35x
18 = x
So I can ride for 18 miles on $8.
Thinking back on this problem, I am proud to say that I had
absolutely no help whatsoever on this particular POW. :) All you had
to do for this one was figure out the equation and the rest was
Page 46 of 58
smooth sailing, but even that wasn't hard. This problem didn't
require much thinking which is good for procrastinators, like me,
who refuse to even look at this problem until late Friday night. :)
Please have more problems like these, thank you!
Student 43
My answer is in the
explanation
-Heather Humpleman, Mr. Compton's Algebra Class, Rancho San Joaquin
Middle School, California, USA
1. 4m = 3.10
7m = 4.15
m = .775
2. 8.025
3. 11miles
Student 44
The Linear Equation is
y = 0.35x + 1.7.
1. First, it asks to found what the equation is so I will first write
the information given in a linear equation which is y = ax + b, here
a is the slope and b is the y-intercept.
First bunch of information became 3.10 = 4a + b
Second bunch became 4.15 = 7a + b
a is the amount charged per mile
Now in order to find a, I do the Linear combination method
4.15 = 7a + b
- 3.10 = 4a + b
-------------------1.05 = 3a
1.05 = 3a
0.35 = a
Now we have found a, we put it in to find b.
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Page 47 of 58
3.10 = (4)(0.35) + b
3.10 = 1.4 + b
1.7 = b
Now that we found a and b, we put it into the other equation to check
4.15 = (7)(0.35) + 1.7
4.15 = 4.15
It is correct so now we create our equation
X is the amount of miles driven
Y is the cost
Y = 0.35X + 1.7
2. We put 12 miles into the equation
Y = (0.35)(12) + 1.7
Y = 4.2 + 1.7
Y = 5.9
The cost for a 12-mile ride is $5.90
3. We put $ 8 into the equation
8 = 0.35X + 1.7
6.3 = 0.35X
18 = X
$8 can take you 18 miles.
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Page 48 of 58
I find this problem to be a review about the slopes, and ways to
solve the linear equations. First, I had to find the slope of the
whole linear equation so I used the linear combination method to
combine the other equations. After I found the slope, I had to find
the y-intercept and then once when I got the y-intercept, the whole
equation came out and the rest of the problem became easier to solve.
I like the problem because it refreshes my memories about slopes.
Extra
Since you are going to give the driver a 10% tip of the cost, you
have to add 10% of the cost into the equation.
This is going to be the new equation
Y = 0.35X + 1.7 + 0.1(0.35X + 1.7)
Now we put $10 into the equation and we solve
10 = 0.35X + 1.7 + 0.1(0.35X + 1.7)
10 = 0.385X + 1.87
8.13 = 0.385X
21.1 = X
Student 45
1. 4x+b=3.20
7x+6=4.15 2. The cost
of a five mile ride
would be $5.90. 3. If
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You can go 21.1 miles with $10 and a 10% tip.
(x=miles/ c=cost/ n=coefficient of x)
To find the linear equation I simply plugged in the numbers into the
Page 49 of 58
you have $8 to spend
on the taxi ride you
could fo 18 miles.
EXTRA: If i plan to tip
the cab driver 10% of
the cost of the ride
and only had $10 to
spend i could
linear function y=mx+b. I therefore came up with the system of
equations of 4x+b=3.10 and 7x+6=4.15.
To find the cost of a twelve-mile ride I plugged 12 into the
equation in place of the coefficient of x which gave me the equation
12x+b=c. To solve this equation i first had to find what b and x
equalled. To do this I substituted into the system of equations and
came up with x equalling 0.35 and b equalling 1.70. I then
substituted these numbers into the 12x+b=c equation and came up with
the twelve-mile ride costing $5.90.
To find out how far you could go if you had only $8 to spend I
substituted 8 into the equation which gave me 8=1.70+0.35n. I then
solved this equation and got n=18. Thus the trip could only be 18
miles with $8 to spend.
Student 46
1. The linear equation
is y=0.35x+1.7 2. The
cost of a 12-mile ride is
$5.90 3. I can go 18
miles Extra: I can go
21.1 miles
EXTRA: To find out how far I could go with only $10 to spend and a
10% tip I substituted into the equation which gave me 9=1.70+0.35n
(the total cost would be $9 if the tip is 10% which is one dollar).
Thus solving this equation gave me n=20.9.
Let y=the cost of the ride and x be the miles you go
Given the taxi rates are based on a linear equation.
y=mx+c, where m is the gradient and c is the constant
1)So, when x=4, y=3.10
3.10=4m+c
c=3.10-4m -----(1)
AND
4.15=7m+c
c=4.15-7m ------(2)
combining (1) and (2):
3.1-4m=4.15-7m
3m=1.05
m=0.35
Thus, y=0.35x+c
by substituiting y=3.1 and x=4 into this equation, we get
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Page 50 of 58
3.1=1.4+c
c=1.7
thus, the linear equation is y=0.35x+1.7
2) when x=12, y=0.35(12)+1.7
y=5.90
Therefore, the cost of a 12-mile ride is $5.90
3) when y= 8, 8=0.35x+1.7
0.35x=6.3
x=18
Thus, i can go 18 miles with $8
Extra: Let your total spending be s
so, s= y*110%
When s=10, y=10/110%
so, 10/110%=0.35x+1.7
0.35x=10/110%-1.7
x=21.1(nearest tenth)
so, i can go 21.1 miles with $10 to spend
Student 47
1. y = (0.35)X + 1.70 2.
5.90 3. 18 Extra : 21.1
1. x = # of miles
y = cost
-> y = mx + b
(4, 3.10)(7, 4.15) -> (4 mile, cost)(7mile, cost)
m (slope) = 4.15-3.10/ 7-4 = 1.05/ 3 = 0.35
y = (0.35)x + b
3.10 = (0.35) * 4 + b -> b = 3.10 - 1.40 = 1.70
-> y= (0.35)x + 1.70
2. x = 12 -> because it said find the cost of a 12 mile ride.
y = (0.35)x + 1.70
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Page 51 of 58
= (0.35)*12 + 1.70 = 5.90
3. 8 = (0.35)x + 1.70
(0.35)x = 6.30 -> x = 6.30/0.35 = 18
Student 48
1. The expression for
the taxi rates in
Lineville is $1.70 +
$.35x 2. The cost of a
12 mile ride is $5.90. 3.
You can go 18 miles on
$8.
Extra : -> i have to pay for the cost of taxi + 10% tip with $10.
when i say y = taxi cost. i can find the taxi cost by using these
equation.
y = cost of taxi
10 = y(1 + 0.1)
y = 10/1.1 = 9.091
9.091 = (0.35)x + 1.70
(0.35)x + 7.39 1
x = 7.391 / 0.35 = 21.1)
-> i have to pay for the cost of taxi + 10% tip with $10. when i say
y = taxi cost. i can find the taxi cost by using these equation.
This problem can be solved by simply subtracting the two equations
needed for this problem. Now, you have the cost of a four mile ride,
and the cost of a seven mile ride. Let's concentrate on the 4 mile
ride first. First I wrote:
x = cost per mile of your taxi ride
With my variable defined, I moved on:
4x = the cost of a four mile ride
However, this does not quite work. The taxi rate does not start at
zero. It is going to be something like: You start with a $1 fee, and
then it's 50 cents per mile. With this, we can complete our equation
y = fee tacked on to the ride
x = Cost per mile of a taxi ride
y + 4x = cost of a four mile ride, which is $3.10
Now that we have the four mile ride finished, we can do the same for
the seven mile ride.
y + 7x = cost of a seven mile ride, which is $4.15
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Page 52 of 58
We have our two equations, so we can solve the problem by simply
subtractimg them. Watch:
y + 7x = $4.15
- y + 4x = $3.10
-------------3x = $1.05
x = $.35
Write the problem
Subtract
Divide 1.05 by 3
So, now we know that the cost per mile in the taxi is 35 cents, we
can plug that in to our problem to find y.
y + 2.45 = $4.15
Plug in .35 for x
y = 1.7
Subtract 2.45 from both sides
Let's check our answer with the other equation
y + 1.4 = $3.10
Plug in .35 for x
y = 1.7
Subtract 1.4 from both sides
In both cases, $1.70 is equal to y. It must be right! Let's put it
together to make an equation!
x = length of ride
$1.70 + $.35x = rate for taxis in Lineville
Using this equation, we can solve the two problems given to us. The
cost of a 12 mile ride is $5.90.
1.70 + .35(12) = cost of a 12 mile ride Plug in 12 for x
$5.90 = cost of a 12 mile ride
Arithmetic
The last problem is, how far can you go on $8? The answer is that
you can go 18 miles on 8 dollars. Look:
x = legnth of ride
1.7 + .35x = 8 dollars
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Write the problem
Page 53 of 58
.35x = 6.3
x = 18
Subtract 1.7 from each side
Divide 6.3 by .35
That's the answer!
Student 49
Our answers were:
1)Our linear equation
is X + MY = C. 2) At 12
mile ride costs $5.90.
3) For $8.00 you can
go 18 miles.
Reflection: This was a very interesting problem. As a consumer, you
need to use this skill to find out how much you really are paying
for something. Using the skills that I used to figure out this
problem can save you hundreds of dollars per year, because you now
know how to see what is a rip off, and what isn't.
1) For our linear equation we got; X + MY + COST. X stands for
the
initial cost, M stands for miles, Y satnds for cost per mile, and C
stands for the total cost of the taxi ride. Our linear equation fits
the needs of the taxi driver or customer so that they can find out
what the ride costs them quickly ans easily.
Then in order to find out what X and Y equal we used linear
combination. Our work looks like this:
X + 7Y = $4.15
__
X + 4Y = $3.10
__________________
3Y = $1.05
____ ______
3
3
Y = $.35
This is how we found what Y equaled. Then to find out what X
equaled
we used substitution. Our work looked like this:
X + 4Y = $3.10
= X + (4 * $.35) = $3.10
= X + $1.40 = $3.10
- $1.40 - $1.40
________________
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X = $1.70
That is how we found our X value.
2) A 12 mile ride costs $5.90. We found this by using
substitution. Our work looked like this:
X + 12Y = C
= $1.70 + ( 12 * $.35) = C
= 1$.70 + $4.20 = C
= $5.90
That is how we found our answer to #2.
3) We found out that you can go 18 miles on $8. We used
substitution to find the answer. Our work looked like this:
$1.70 + M.35 = $8.00
-$1.70
-$1.70
____________________
M.35 = $6.30
____ _____
.35
.35
M = 18
That is how we found the answer to the third problem.
Student 50
1. y = 0.35x + 1.7 is the
linear equation that
fits the given
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Reflection: This problem confused us at first. Then we realized
that
we had to use both equations that had been given in the
problem.
Then this problen didn't take long at all. We solved it easily. This
also put a new perspective on linear equations.
Facts:
4-mile ride = $3.10
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information. 2. The
cost of a 12-mile ride is
$5.90 3. A person can
go up to 18 miles if
he/she only has $8.00
to spend.
7-mile ride = $4.15
linear equation is y = mx + b
•
•
•
•
y =
m=
x =
b =
total cost of ride
cost of each mile rode
number of miles rode
extra charge
Keeping these facts in mind, I formulated the following equations:
3.10 = m4 + b
4.15
= m7 + b
1.) In order to figure out the linear equation I first had to figure
the m value. I didn’t have to worry about the b value right now, so
I set aside b to the other side of the equation:
b = 3.10 - m4
b = 4.15 - m7
Now to solve for m:
3.1 - m4 = 4.15 + m7
- m4 = 1.05 - m7
m3 = 1.05
m = 0.35
My linear equation so far is y = 0.35x + b. To solve for b, I
plugged in the y and x values from the given clues:
Solve for b:
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3.10 = 4 (0.35) + b
3.10 = 1.4 + b
1.7 = b
The linear equation is y = 0.35x + 1.7
2.) Solve for 12 miles:
y = 0.35 (12) + 1.7
y = 4.2 + 1.7
y = 5.9
The cost of a 12-mile ride is $5.90.
3.) Solve for $8 as the total cost (plug in 8 for y):
8 = 0.35x + 1.7
6.3 = 0.35x
18 = x
You can travel 18 miles with only $8.00.
Reflection:
This was the easiest POW by far. Understanding what each of
the variables in the linear equation stood for was the key to
solving this problem. Laying out my facts was the trickiest part of
the problem. It was smooth sailing once I got over that step. As
always, careful substitution played a huge role in solving the
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problem. Solving problems with the linear equation will always be a
part of everyday life and this problem proves that fact.
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