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L13: Solve Linear Equations with Rational Coefficients
116
Solve Linear Equations with Rational
Coefficients
Part 1: IntroductionLesson 13
You’ve learned how to solve linear equations with variables on one side of the
equation. In this lesson, you’ll learn how to solve linear equations with variables on
both sides of the equation. Take a look at this problem.
The square and equilateral triangle shown have
x 1 3 3x 2 1
the same perimeter. What is the value of x?
Explore It
Use the math you already know to solve this problem.
The perimeter of the square is 4(x 1 3), and the perimeter of the triangle is 3(3x 2 1).
Because the perimeters are the same, 4(x 1 3) 5 3(3x 2 1). Try substituting 3 for x in the
equation. Do you get a true statement? Explain.
Use the distributive property to transform 4(x 1 3) 5 3(3x 2 1) into a simpler form
without parentheses.
Substitute 3 for x in your new equation. Do you get a true statement? Explain.
You can transform 4x 1 12 5 9x 2 3 again into a simpler form by subtracting 4x from
both sides to get 12 5 5x 2 3. If you substitute 3 for x in 12 5 5x 2 3, do you still get a
true statement?
You can continue to transform 12 5 5x 2 3 into a simpler form by adding 3 to both sides
to get 15 5 5x. When x 5 3, do you get a true statement?
Finally, divide both sides of 15 5 5x by 5. What is the result?
Develop Skills and Strategies
CCLS
8.EE.C.7b
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L13: Solve Linear Equations with Rational Coefficients
Lesson 13Part 1: Introduction
Find Out More
In Explore It, you transformed the original equation into simpler and simpler forms. Each
time, substituting 3 for x resulted in a true statement. The transformations don’t change the
solution to the equation.
You transform equations to get to the form x 5 a because that gives you the solution to the
equation you started with. You can add and subtract variables from both sides of the
equation just as you do with constants.
Look at a step-by-step solution of 4(x 1 3) 5 3(3x 2 1).
4(x 1 3) 5 3(3x 2 1)
4x 1 12 5 9x 2 3
12 5 5x 2 3
15 5 5x
3 5 x
There is often more than one way to solve an equation.
4(x 1 3) 5 3(3x 2 1)
4x 1 12 5 9x 2 3
2 5x 1 12 5 2 3
2 5x 5 2 15
x 5 3
Reflect
1
How do you solve multi-step equations that have variables on both sides?
Apply the distributive property.
Subtract 4x from both sides so that the
variable occurs on just one side.
Add 3 to both sides.
Divide both sides by 5.
Apply the distributive property.
Subtract 9x from both sides so that the
variable occurs on just one side.
Subtract 12 from both sides.
Divide both sides by
2 5.
2 4x 2 4x
1 3 1 3
5 5
2 9x 2 9x
2 12 2 12
2 5 2 5
Lesson 13
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L13: Solve Linear Equations with Rational Coefficients
118
Part 2: Modeled Instruction
Read the problem below. Then explore different ways to solve an equation.
On a math quiz, Elise and Kaitlyn solved the equation
1
··
2
(5x 2 6) 5 3x in different
ways, but each student arrived at the correct answer. Describe the steps that each
student took to solve the equation. An explanation of the first step for each method
has beenprovided.
Solve It
Elise solved the problem in this way.
1
··
2
(5x 2 6) 5 3x
5
··
2
x 2 3 5 3x
2
5
··
2
x 2
5
··
2
x
2 3 5
1
··
2
x
1
··
2
1
··
2
2 6 5 x
Solve It
Kaitlyn solved the problem in this way.
1
··
2
(5x 2 6) 5 3x
2 •
1
··
2
(5x 2 6) 5 2 • 3x
5x 2 6 5 6 x
2 6 5 x
Step 1 Apply the distributive property.
Step 2 Subtract
5
··
2
x from each side.
Step 3 Divide both sides by
1
··
2
.
Step 1 Multiply both sides by 2.
Step 2 Subtract 5x from both sides.
2 5x 2 5x
Lesson 13
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L13: Solve Linear Equations with Rational Coefficients
Part 2: Guided Instruction
Connect It
Now you will analyze how each student solved the equation.
2
Look at Elise’s solution method. She took three steps to solve the equation. Describe
Step 2.
Why do you think Elise took that step?
3
Describe Step 3 in Elise’s solution.
Could Elise have used a different step? Explain.
4
Look at Kaitlyn’s solution method. Describe Step 2 in her solution.
Why do you think she took that step?
5
Which method do you prefer? Explain your thinking.
6
Explain how to check the solution to an equation. Then show how to check the solution
to the equation on the previous page.
Try It
Use what you learned about different ways to solve linear equations. Show your work.
7
Solve the equation and check your solution: 10 5
1
··
3
(x 2 15).
Student Model
Lesson 13
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L13: Solve Linear Equations with Rational Coefficients
120
Study the student model below. Then solve problems 8–10.
Solve the following equation for r.
16.5 1 1.5r 5 12 1 2r
Look at how you can show your work.
Solution:
8
One fifth of a number plus three times the number is equal to twice the
number plus 42. What is the number?
Show your work.
Solution:
How can you check your
answer?
Pair/Share
What equation can you
write to solve the problem?
Can you solve this
equation in another way?
Pair/Share
The student multiplied
each side of the equation
by 10 to transform the
equation into one without
decimals.
Part 3: Guided Practice
9 5 r
16.5 1 1.5r 5 12 1 2r
10(16.5 1 1.5r) 5 10(12 1 2r)
165 1 15r 5 120 1 20r
2 15r 2 15r
165 5 120 1 5r
2120 2120
45 5 5r
5 5
Lesson 13
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L13: Solve Linear Equations with Rational Coefficients
How would you help
Haley understand
her error?
Pair/Share
Justify the steps you took
to solve the equation.
Pair/Share
What properties and
operations can you use
to simplify both sides of
an equation?
Can you multiply both
sides of the equation by a
number to get a simpler
equation without
fractions?
9
Show two different ways to solve
1
··
4
x 2 5 5
3
··
4
x 2 12.
Show your work.
Solution:
10
Which equation has the same solution as
1
··
2
(6 2 x) 1 3x 5
1
··
2
x 2 8?
Circle the letter of the correct answer.
A 3 1 2x 5
1
··
2
x 2 8
B 6 2 x 1 3x 5 x 2 16
C 3 1
5
··
2
x 5
1
··
2
x 2 8
D 6 2 x 1 3x 5 x 2 8
Haley chose A as the correct answer. How did she get that answer?
Part 3: Guided Practice
