Mathematical Proof – Linear Equation
Justify each step for solving the equations below. Use the following for your justification:
- Distributive Property
- Combine like terms
- Addition Property of Equality
- Subtraction Property of Equality
- Division Property of Equality
- Multiplication Property of Equality
1) 4𝑥 = 12𝑥 + 32
–8𝑥 = 32
2) 28 + 12𝑥 = 8𝑥 – 4
Given
Given
28 + 4x = –4
Subtraction Property of Equality
𝑥 = –4
- Given
Subtraction Property of Equality
4𝑥 = –32
Division Property of Equality
Subtraction Property of Equality
𝑥 = −8
3) 60𝑥 + 153 = 9𝑥 + 51
51𝑥 + 153 = 51
51𝑥 = –102
𝑥 = −2
5) –3(𝑥 + 2) = 16 – 𝑥
−3𝑥 − 6 = 16 − 𝑥
−6 = 16 + 2𝑥
Given
4) –4𝑥 + 10 = –5𝑥 + 18
Subtraction Property of Equality
𝑥 + 10 = 18
Subtraction Property of Equality
𝑥=8
Given
Distributive Property
Subtraction Property of Equality
−𝑥 − 18 + 16𝑥 = 12
15𝑥 − 18 = 12
Addition Property of Equality
−11 = 𝑥
Division Property of Equality
Given
6𝑥 − 36 = 16𝑥 − 7𝑥
Distributive Property
6𝑥 − 36 = 9𝑥
Combine Like Terms
𝑥 = −12
Addition Property of Equality
6) –𝑥 – 2(9 – 8𝑥) = 12
Subtraction Property of Equality
−36 = 3𝑥
Given
Division Property of Equality
−22 = 2𝑥
7) 6(𝑥 – 6) = 𝑥(16 – 7)
Division Property of Equality
𝑥=2
8)
Subtraction Property of Equality
Division Property of Equality
15𝑥 = 30
x
4
+ 10 = 2
𝑥
= −8
4
𝑥 = −32
Given
Distributive Property
Combine Like Terms
Addition Property of Equality
Division Property of Equality
Given
Subtraction Property of Equality
Multiplication Property of Equality