Solve Multi-Step Equations
Involving Rational Numbers
Jen Kershaw
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Printed: July 16, 2013
AUTHOR
Jen Kershaw
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C ONCEPT
Concept 1. Solve Multi-Step Equations Involving Rational Numbers
1
Solve Multi-Step Equations
Involving Rational Numbers
Here you’ll learn to solve multi-step equations involving rational numbers.
Jose has been playing the French horn for many years and until now, everything has been easy. Now Mrs. Kline, the
band director has assigned him a new piece of music to work on and it is very tricky. Jose has been practicing the
new piece.
His best rehearsal was on Saturday, when he practiced for 90 minutes. On Sunday, he had a birthday party to go to,
so he did not practice as long. On Monday, he had a math test to study for and so he practiced for half as long as he
did on Monday.
When Jose went to band practice on Tuesday afternoon, he struggled through the piece.
“How long did you practice?” Mrs. Kline asked him.
“Well, from Saturday to Tuesday I practiced a total of 3 hours,” Jose said.
If this is true, how long did Jose practice on Monday and Tuesday? You will need to write an equation and
solve it to figure out the answer to this problem. Jose needs to practice his French horn a bit more and you
will need to use the information taught in this Concept to help you figure out each dilemma.
Guidance
Rational numbers include integers, fractions, and terminating decimals. Some equations may require you to work
with a combination of these kinds of numbers. If you know how to solve an equation, you can apply the same rules
when you work with rational numbers.
Take a look at this dilemma.
2
b
Solve for b: −6 1 − 12
=3
b
This problem involves two different kinds of rational numbers: integers (-6 and 1) and fractions 12
and 23 .
You will need to know how to compute with fractions as well as how to compute with integers in order to solve
this.
1
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Apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the
parentheses by -6 and then subtract those products.
b
=
−6 1 −
12
b
(−6 × 1) − −6 ×
=
12
−6
1
−6 −
× b =
1
12
−6
−6 −
b =
12
6
−6 − − b =
12
6
b =
−6 +
12
2
3
2
3
2
3
2
3
2
3
2
3
6
You may recognize immediately that the variable term, 12
b, could be simplified as 12 b or b2 . You can wait until the
problem is finished before simplifying, but if you recognize this fact, it makes sense to simplify that now. It will
only make the computation easier, so simplify the variable term as 12 b.
6
b=
12
1
−6 + b =
2
−6 +
2
3
2
3
Now, we can solve as we would solve any two-step equation. To get 12 b by itself on one side of the equation, we
can subtract -6 from both sides.
1
2
−6 + b =
2
3
1
2
−6 − (−6) + b = − (−6)
2
3
1
2
−6 + 6 + b = + 6
2
3
1
2
0+ b = 6
2
3
1
2
b=6
2
3
To get b by itself, you will need to divide each side of the equation by 12 . Remember, that is the same as multiplying
each side by 21 . Also, keep in mind that you will need to rewrite the mixed number 6 23 as an improper fraction 20
3
before multiplying by 21 .
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Concept 1. Solve Multi-Step Equations Involving Rational Numbers
1
2
b=6
2
3
1
2
2 2
b× = 6 ×
2
1
3 1
2A 20 2
1A
b× =
×
2A
1A
3
1
40
1b =
3
1
b = 13
3
The value of b is 13 13 .
Now, let’s solve an algebraic equation that includes both decimals and fractions.
3
=
Solve for k: 0.4k + 0.2k + 10
9
10 .
First, add the like terms 0.4k and 0.2k on the left side of the equation.
3
9
=
10 10
3
9
0.6k +
=
10 10
0.4k + 0.2k +
The next step is to isolate the term with the variable, 0.6k, on one side of the equation. We can do this by
3
subtracting 10
from both sides of the equation.
3
9
=
10 10
3
9
3
3
=
−
0.6k + −
10 10 10 10
6
0.6k + 0 =
10
6
0.6k =
10
0.6k +
Since 0.6k means 0.6 × k, we should divide each side of the equation by 0.6 to get the k by itself on one side of
6
, by a decimal, 0.6. To do this, you will need to convert both
the equation. This will involve dividing a fraction, 10
6
6
numbers to the same form. One way to do this would be to convert the fraction 10
to a decimal. 10
is read as “six
6
tenths,” so the decimal form of 10 is 0.6.
6
10
0.6k = 0.6
0.6k 0.6
=
0.6
0.6
1k = 1
0.6k =
k=1
The value of k is 1.
3
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Example A
8.7n − 3.2n + 4.5 = 37.5
Solution: n = 6
Example B
x
.9
= −72
Solution: x = −64.8
Example C
17x − 22.3x + 4 = −33.1
Solution: x = 7
Now let’s go back to the dilemma at the beginning of the Concept.
First, write an equation to show what you know and what you don’t know.
Saturday = 90 minutes
Monday = t − missing time
Tuesday = 12 t − half the time of Monday
Total time = 3 hours
90 + t + 12 t = 3 hours
First, convert hours to minutes.
90 + t + 12 t = 180 minutes
Now we can solve the equation.
Monday’s time = 60 minutes
Tuesday’s time = 30 minutes
Vocabulary
Integer
the set of whole numbers and their opposites.
Rational Numbers
a set of numbers that includes integers, decimals, fractions, terminating and repeating decimals. These
numbers can be written in fraction form.
Fraction
a part of a whole written using a numerator and a denominator.
Decimal
a part of a whole written using place value and a decimal point.
Repeating Decimal
a decimal where the digits repeat in a pattern and eventually end.
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Concept 1. Solve Multi-Step Equations Involving Rational Numbers
Terminating Decimal
a decimal where the digits eventually end, but where numbers do not repeat in a pattern.
Guided Practice
Here is one for you to try on your own.
For a long-distance call, Guillermo’s phone company charges $0.10 for the first minute and $0.05 for each minute
after that. Guillermo was charged $1.00 for a long distance call he made last Friday.
a. Write an algebraic equation that could be used to represent m, the length in minutes of Guillermo’s $1.00 longdistance call.
b. Determine how many minutes his $1.00 long-distance call lasted.
Solution
Consider part a first.
You know that the phone company charges $0.10 for the first minute and $0.05 for each minute after that. How
could you represent that? If the company charged $0.05 for each minute the call lasted, you could represent that as
0.05 × m. However, the company charges $0.10 for the first minute and $0.05 for each minute after that first minute.
So, a 1-minute call will cost: $0.10 + ($0.05 × 0) = $0.10 + $0.00 = $0.10.
A 2-minute call will cost: $0.10 + ($0.05 × 1) = $0.10 + $0.05 = $0.15.
A 3-minute call will cost: $0.10 + ($0.05 × 2) = $0.10 + $0.10 = $0.20.
Notice that the number you multiply by $0.05 is always 1 less than the length of the call, in minutes. If m represents
the length of a call in minutes, then this could be represented as: $0.10 + $0.05 × (m − 1).
Write an equation that could be used to represent the cost of Guillermo’s $1.00 call.
(cost of first minute) + (cost of each minute after first minute) = (total cost)
↓
↓
0.10
+
↓
0.05(m − 1)
↓
↓
=
1.00
So, the equation 0.10 + 0.05(m − 1) = 1.00 represents the number of minutes that Guillermo’s $1.00 phone call
lasted.
Next, consider part b.
To find the length of the $1.00 call in minutes, solve the equation for m. First, apply the distributive property to the
right side of the equation.
0.10 + 0.05(m − 1) = 1.00
0.10 + (0.05 × m) − (0.05 × 1) = 1.00
0.10 + 0.05m − 0.05 = 1.00
Use the commutative property to rearrange the terms being added so it is easier to see how to combine the like terms.
Then combine the like terms.
5
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(0.10 + 0.05m) − 0.05 = 1.00
(0.05m + 0.10) − 0.05 = 1.00
0.05m + (0.10 − 0.05) = 1.00
0.05m + 0.05 = 1.00
Now, solve as you would solve any two-step equation. First, subtract 0.05 from both sides of the equation.
0.05m + 0.05 = 1.00
0.05m + 0.05 − 0.05 = 1.00 − 0.05
0.05m + 0 = 0.95
0.05m = 0.95
Next, divide both sides of the equation by 0.05.
0.05m = 0.95
0.05m 0.95
=
0.05
0.05
1m = 19
m = 19
The value of m is 19, so the $1.00 call lasted for 19 minutes.
Video Review
MEDIA
Click image to the left for more content.
KhanAcademySolvingLinear Equations 4
Practice
Directions: Solve each equation to find the value of the variable.
1.
2.
3.
4.
5.
6
7n − 3.2n + 6.5 = 17.9
0.2(3 + p) = −5.6
s + 35 + 15 = 1 25
j + 57 − 17 = 9 47
3
1
1
4 g− 2 = 8
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6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Concept 1. Solve Multi-Step Equations Involving Rational Numbers
−2 1 − 4a = 18
0.09y − 0.08y = .005
.36x + 2.55x = −8.55
1
1
3y+ 3y = 8
1
1
2
4x+ 3 = 3
1
2 x = 18
.9x = 56
.6x + 1 = 19
1
4 x + 2 = 19
9.05x = 27.15
7