Example 1
During a weekend, Mary watched 8 television shows. Short shows were 30 minutes and long shows were
60 minutes. If she spent 5 hours (300 minutes) watching TV, how many short TV shows did she watch? How
many long TV shows did she watch?
What are you asked to determine? Choose a variable to represent each unknown.
Number of long TV shows ( )
Number of short TV shows ( )
What information are you given?
8 shows were watched
Short shows are 30 minutes
Long shows are 60 minutes
5 hours (300 minutes) were spent watching shows
Stop and think about what this means.
Table Method
Based on the following possibilities for the number of Long Shows, how many Short Shows would have to be watched?
Long Shows
Short Shows
0
1
2
3
4
5
6
7
8
Why stop at 8 Long Shows?
Now take the possible # of Long Shows and Short Shows and determine the corresponding amount of time.
Time Spent
Time Spent
Total Time
Watching
Watching
Spent
Long Shows
Short Shows
Watching TV
0*60=
1
2
3
4
5
6
7
8
Where can you find the answer in your table? How do you know that’s the answer?
Equation Method
Since we have two unknowns, we need two equations.
Remember, those two unknowns are:
Number of long TV shows ( )
Number of short TV shows ( )
We know the total number of long and short shows that were watched (8).
We know the total time spent watching TV shows (300 minutes)
The time spent watching a show can be determined by multiplying the number of shows by the length of
the show.
Write and solve the system of equations.
We will use substitution since we have at least one variable with a coefficient of 1
{
Solve one equation for a variable
Substitute the resulting expression into the
other equation
Solve. (Distribute, add like terms, collect
constants, isolate the variable)
Substitute the value into the other equation
and solve.
Write as an ordered pair
Mary watched 2 long TV shows and 6 short TV shows.
Practice – Use either the Table Method or Equation Method
1. A bakery sells regular pies that are 9 inches in diameter and mini-pies that are 4 inches in diameter. The shelf
they out the pies on is 72 inches long. If there are 13 pies on the shelf total, how many of each type of pie are
there?
a. What is the problem asking you to find? These are your unknowns or variables.
b. What does the problem tell you? Remember, you’ll have two tables or two equations.
c. Hint: One table/equation will be about the total number of pies and the other will be about length they
take up on the shelf.
2. Kobe Bryant made a total of 20 shots in a game. Some of his shots were 2-pointers and the others were 3pointers, scoring a total of 48 points. How many of each type of shot did he make?
Now do problem #2 from your quiz.
Example 2
You went to the grocery store to buy vegetables. You spent a total of $12.50 on broccoli and asparagus. Broccoli
costs $0.50 per pound and asparagus costs $1.50 per pound. If you bought twice as much broccoli as asparagus,
how many pounds of each vegetable did you buy?
What are you asked to determine? Choose a variable to represent each unknown.
Pounds of Broccoli (x)
Pounds of Asparagus (y)
What information are you given?
Total spent: $12.50
Broccoli costs $0.50 per pound
Asparagus costs $1.50 per pound
Bought twice as many pounds of broccoli as asparagus
Table Method
Think about what this means: you bought twice as much broccoli as asparagus.
Did you buy more broccoli or more asparagus? (broccoli)
If we know the amount of asparagus, how can we determine the amount of broccoli? (twice as much means
multiply by 2)
Fill in some possible amounts of broccoli based on some possible amounts of asparagus. (Note: this not all of the possible
combinations, you might need to extend the table).
Asparagus Broccoli
1
2
3
4
5
Asparagus Broccoli
6
7
8
9
10
y
Now take the possible combinations of the amounts of each vegetable and determine the total cost of each and the total
cost of both together
Money spent on
Money spent on
Total spent on
asparagus
broccoli
both
0*$1.50=
1
2
3
4
5
6
7
8
How many pounds of each vegetable were purchased?
Equation Method
Since we have two unknowns, we need two equations.
Remember, those two unknowns are:
Pounds of Broccoli (x)
Pounds of Asparagus (y)
We know the total cost of the vegetables ($12.50) and the cost per pound of each vegetable ($0.50 per
pound broccoli and $1.50 per pound of asparagus)
We know that we bought twice as much broccoli as asparagus.
This means that if we want to know the amount of broccoli, we would multiply the amount of asparagus,
by 2 to get the amount of broccoli (see table method if unsure of why)
Write and solve the system of equations.
We will use substitution since we have at least one variable (x) already solved for.
{
One equation already solved for a variable
Substitute the resulting expression into the
other equation
Solve. (Distribute, add like terms, collect
constants, isolate the variable)
Substitute the value into the other equation
and solve.
Write as an ordered pair
What does the solution mean in the context of this problem? Hint: look back at what x and y represent.
Example 3
Let’s change the previous problem just a little bit. You went to the grocery store to buy vegetables. You spent a
total of $17.50 on broccoli and asparagus. Broccoli costs $0.50 per pound and asparagus costs $1.50 per pound. If
you bought 3 fewer pounds of asparagus than broccoli, how many pounds of each vegetable did you buy?
We’re trying to determine the same things as before, so we’ll use the same variables.
Think about what this means: you bought three fewer pounds of asparagus than broccoli
Did you buy more broccoli or more asparagus? (broccoli)
If we know the amount of broccoli, how can we determine the amount of asparagus? (three fewer means
subtract 3)
Fill in some possible amounts of asparagus based on some possible amounts of broccoli. (Note: this not all of the possible
combinations, you might need to extend the table).
Broccoli Asparagus
1
2
3
4
5
Broccoli Asparagus
6
7
8
9
10
x
Now take the possible combinations of the amounts of each vegetable and determine the total cost of each and the total
cost of both together
Money spent on
Money spent on
Total spent on
broccoli
asparagus
both
0*$1.50=
1
2
3
4
5
6
7
8
How many pounds of each vegetable were purchased?
Equation Method
Remember, those two unknowns are:
Pounds of Broccoli (x)
Pounds of Asparagus (y)
We know the total cost of the vegetables ($17.50) and the cost per pound of each vegetable ($0.50 per
pound broccoli and $1.50 per pound of asparagus)
We know that we bought three pounds less of asparagus than broccoli.
This means that if we want to know the amount of asparagus, we would subtract 3 from the amount of
broccoli.
Then solve using substitution.
Practice Problems
1. 84 people attended a baseball game. Everyone there was a fan of either the home team or the away team. The
number of home team fans was 3 times more than the number of away team fans. How many fans for each team
were there?
2. A group of adults and kids went to see a movie. Tickets cost $8.00 each for adults and $4.50 each for kids, and the
group paid $72.50 in total. There were 5 fewer adults than kids in the group.
Now do #4 on the quiz.
Example 4
You’ve gone to a produce stand to by some fresh fruit. You notice that the person in front of you buys 5
apples and 4 oranges for $9.60. You buy 3 apples and 8 oranges for $10.80. What is the cost of each apple
and orange?
What are you asked to determine? Choose a variable to represent each unknown.
Cost of one apple ( )
Cost of one orange ( )
What information are you given?
The cost of 5 apples and 4 oranges is $9.60
The cost of 3 apples and 8 oranges is $10.80
Since we have two unknowns, we need two equations.
We know the cost for 5 apples and 4 oranges ($9.60)
The total cost is determined by multiplying the cost of each piece of fruit by the number of pieces bought.
We know the cost for 3 apples and 8 oranges ($10.80)
The total cost is determined by multiplying the cost of each piece of fruit by the number of pieces bought.
Write and solve the system of equations.
{
We can either use substitution or elimination.
Substitution
Elimination
Solve one equation for a
variable
(
)
{
Substitute the resulting
expression into the other
equation
Solve. (Distribute, add
like terms, collect
constants, isolate the
variable)
Substitute the value into
the other equation and
solve.
Write as an ordered pair
Check:
Write as an ordered pair
What does the solution represent?
Like terms are already
aligned. Multiply the first
equation by -2 so will
be eliminated.
Eliminate one of the
variables by adding the
“new” equation to the
unchanged equation.
Solve.
Substitute the value into
one of the original
equations and solve.
Example 5
Sally has $20 in her account and is adding $3.50 to her account each week. Joe has $172 in his account, but needs
to pay his parents back $6 per week. After how many weeks will they have the same amount of money in their
account? How much money will that be?
What are you asked to determine? Choose a variable to represent each unknown.
How many weeks until they have the same amount of money ( )
Money in account at the time when the accounts have the same amount. ( )
What information are you given?
Sally already has $20 and is adding $3.50 per week
Joe already has $172 and is taking out $6 per week
Table Method
Weeks
Sally’s Account
Joe’s Account
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20
172
Equation Method
We’re going to have an equation for Sally and an
equation for Joe.
Sally’s equation:
Joe’s equation:
Both equations are
solved for y
Set the two equations
equal to each other.
Solve. (Distribute, add
like terms, collect
constants, isolate the
variable)
What are you looking for in your table to determine the
solution?
Substitute the value into
one equation and solve.
Write as an ordered pair
Check:
What is the solution and what does it represent?
Practice
1. House-Painting Company A charges $376 plus $12 per hour. Company B charges $280 plus $15 per hour. How
long is the job for which both companies will charge the same amount? What will that cost be?
2. Justin and Tyson are beginning an exercise program to train for football season. Justin weighs 150 lb and hopes
to gain 2 lb per week. Tyson weighs 195 lb and hopes to lose 1 lb per week. If the plan works, in how many
weeks will the boys weigh the same amount? What will that weight be?
Now do #1 on your quiz.