LE
SS
O
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9
Solving Linear Equations in
One Variable
UNDERSTAND To solve a linear equation in one variable, simplify the equation. Sometimes you may have to use the distributive property and combine
like terms. Then isolate the variable on one side of the equation and
solve for the variable. The resulting equation will tell you whether
there is one solution, no solutions, or infinitely many solutions.
Solve: 4(x 1 2) 2 2x 5 4x 2 2
Apply the distributive property. 4(x 1 2) 2 2x 5 4x 2 2
(4  x) 1 (4  2) 2 2x 5 4x 2 2
4x 1 8 2 2x 5 4x 2 2
Combine like terms. 4x 1 8 2 2x 5 4x 2 2 4x and 22x are like terms. 2x 1 8 5 4x 2 2
Apply properties of equality so that the variable is on just one side of the equation.
2x 1 8 5 4x 2 2 Subtract 2x from both sides to remove x from the left side. 2x 2 2x 1 8 5 4x 2 2 2 2x
8 5 2x 2 2
Apply properties of equality to solve for x. 8 5 2x 2 2
10 5 2x
10 4 2 5 2x 4 2
55x
Check the solution by substituting into the original equation. 4(x 1 2) 2 2x 5 4x 2 2
4(5 1 2) 2 2(5) 0 4(5) 2 2
4(7) 2 2(5) 0 4(5) 2 2
4(7) 2 10 0 20 2 2
52 28 2 10 0 18
18 5 18 ✓ The solution checks.
Domain 2: Expressions and Equations
Duplicating any part of this book is prohibited by law.
Connect
Solve: 5(2x 2 4) 5 3(3x 2 6) 1 x 2 2
1
Apply the distributive property and combine like terms. 5(2x 2 4) 5 3(3x 2 6) 1 x 2 2
(5  2x) 1 (5  24) 5 (3  3x) 1 (3  26) 1 x 2 2
2
10x 2 20 5 9x 2 18 1 x 2 2
10x 2 20 5 10x 2 20
Apply properties of equality to solve for x.
10x 2 20 5 10x 2 20
10x 2 20 1 20 5 10x 2 20 1 20
10x 5 10x
___
​ ___
10  ​5 ​  10  ​
x5x
10x
10x
The equation is true for all values of x. If you substitute any value for x in the
original equation, it will result in a true
statement. ▸ There are infinitely many solutions.
Solve: 2(x 1 4) 1 3 5 2x 1 6
1
Apply the distributive property and
combine like terms. 2(x 1 4) 1 3 5 2x 1 6
Duplicating any part of this book is prohibited by law.
(2  x) 1 (2  4) 1 3 5 2x 1 6
2x 1 8 1 3 5 2x 1 6
2x 1 11 5 2x 1 6
M O DE L
Write an equation that has infinitely
many solutions and an equation that
has no solution. What must be true
about the variable terms on each side
of the equations?
2
Apply properties of equality.
2x 1 11 5 2x 1 6
2x 2 2x 1 11 5 2x 2 2x 1 6
11 5 6
The equation results in a false statement. 11  6. If you substitute any value for x in
the original equation, it will give a false
statement. ▸ There are no solutions for x.
Lesson 9: Solving Linear Equations in One Variable 53
Practice
Solve each equation for x.
1.
3x 1 1 5 4x 2 2
2.
x5
5(x 2 6) 2 2 5 2x 2 5
x5
REMEMBER Apply the
distributive property before
you isolate the variable.
3.
7x 1 12 5 2(x 1 6)
4.
x5
3(x 2 4) 1 6 5 5(x 2 1) 1 1
x5
5.
3(x 1 4) 2 2 5 2(2x 1 5) 2 x
6.
2x 1 1 2 3x 1 5 5 3(x 1 10)
7.
3(x 2 2) 1 1 5 2(x 2 4) 1 x 1 13
8.
0.1(5x 1 20) 2 5 5 0.25(2x 1 8)
9.
4(2.5x 2 2) 5 2(5x 2 5) 1 2
1
10. __
​ 2 ​(  x 2 6) 1 1 5 2(x 2 10) 2 3
54 Domain 2: Expressions and Equations
Duplicating any part of this book is prohibited by law.
Solve each equation for x. If there are infinitely many solutions, write “infinitely many
solutions.” If there is no solution, write “no solution.”
Complete the steps to solve each equation.
 1

4
11. 2​ __
​ 3 ​ x 1 4  ​2 1 5 __
​ 3 ​x 2 3 1 x
12. 4.5(x 2 2) 1 1.5x 5 2(3x 2 4) 2 1
Apply the distributive property:
Apply the distributive property:
Combine like terms:
Combine like terms:
Isolate and solve for x:
Isolate and solve for x:
Check your answer:
Interpret the answer:
Choose the best answer.
13. What is the solution to the following?
4(x 2 1) 2 3x 5 22x 2 4 1 3x
14. Which equation has exactly one
solution?
A. x 5 24
1
1
A. ​ __
  1 13 1 x 5 2​ __   ​(x 1 40) 1 4
5 ​x
10
B. x 5 0
B. 4x 2 4 1 x 5 2(3x 2 2) 2 x
C. no solution
C. 2(4x 1 5) 2 10 5 4(2x 2 3) 1 12
D. infinite solutions
D. 2.5(2x 2 4) 5 4(x 2 4) 1 x
Solve.
Duplicating any part of this book is prohibited by law.
15.
EXPLAIN If an equation in one variable
contains a variable term on both
sides of the equals sign, explain what
steps you need to take to solve for the
variable.
16.
EXPLAIN In your own words, explain
what it means when a solution to an
equation in one variable results in an
inequality, such as 3  4. Lesson 9: Solving Linear Equations in One Variable 55