Linear Equations Worksheets with Answer Key - Free Printable

Here are the step-by-step solutions for each equation on the worksheet.

1) $8(4a - 4) - 1 = 14 - 8a$
* Distribute the 8 into the parentheses: $32a - 32 - 1 = 14 - 8a$
* Combine like terms on the left side: $32a - 33 = 14 - 8a$
* Add $8a$ to both sides: $40a - 33 = 14$
* Add 33 to both sides: $40a = 47$
* Divide by 40: $a = \frac{47}{40}$ (or $1.175$)

2) $4(-8x + 5) = -30x - 26$
* Distribute the 4 into the parentheses: $-32x + 20 = -30x - 26$
* Add $32x$ to both sides: $20 = 2x - 26$
* Add 26 to both sides: $46 = 2x$
* Divide by 2: $x = 23$

3) $27 = 46 + 3x - x$
* Combine the $x$ terms on the right side ($3x - x = 2x$): $27 = 46 + 2x$
* Subtract 46 from both sides: $-19 = 2x$
* Divide by 2: $x = -\frac{19}{2}$ (or $-9.5$)

4) $38 + 7k = 8(k + 4)$
* Distribute the 8 into the parentheses: $38 + 7k = 8k + 32$
* Subtract $7k$ from both sides: $38 = k + 32$
* Subtract 32 from both sides: $6 = k$
* So, $k = 6$

5) $-2(4 - x) = 6(x + 4) + 4x$
* Distribute the numbers into the parentheses: $-8 + 2x = 6x + 24 + 4x$
* Combine like terms on the right side ($6x + 4x = 10x$): $-8 + 2x = 10x + 24$
* Subtract $2x$ from both sides: $-8 = 8x + 24$
* Subtract 24 from both sides: $-32 = 8x$
* Divide by 8: $x = -4$

6) $0.05z + 0.2 = 0.15z - 10.5$
* Subtract $0.05z$ from both sides: $0.2 = 0.10z - 10.5$
* Add 10.5 to both sides: $10.7 = 0.10z$
* Divide by 0.10: $z = 107$

7) $4(6 + 4x) = 0$
* Distribute the 4: $24 + 16x = 0$
* Subtract 24 from both sides: $16x = -24$
* Divide by 16: $x = -\frac{24}{16}$
* Simplify the fraction (divide top and bottom by 8): $x = -\frac{3}{2}$ (or $-1.5$)

8) $\frac{x}{8} - \frac{x}{9} = 1$
* Find a common denominator for the fractions, which is 72.
* Rewrite the fractions: $\frac{9x}{72} - \frac{8x}{72} = 1$
* Subtract the numerators: $\frac{1x}{72} = 1$
* Multiply both sides by 72: $x = 72$

Final Answer:
1) $a = \frac{47}{40}$ or $1.175$
2) $x = 23$
3) $x = -9.5$
4) $k = 6$
5) $x = -4$
6) $z = 107$
7) $x = -1.5$
8) $x = 72$