Linear Equations Worksheet for solving unknown variables, featuring ten algebraic equations.

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

1) $-\frac{9}{5}x = -45$
* To isolate $x$, multiply both sides by the reciprocal of $-\frac{9}{5}$, which is $-\frac{5}{9}$.
* $x = -45 \cdot (-\frac{5}{9})$
* $x = \frac{-45 \cdot -5}{9}$
* $x = \frac{225}{9}$
* $x = 25$

2) $\frac{x}{3} - \frac{x}{5} = 2$
* Find a common denominator for the fractions, which is 15.
* Rewrite the equation: $\frac{5x}{15} - \frac{3x}{15} = 2$
* Combine the numerators: $\frac{2x}{15} = 2$
* Multiply both sides by 15: $2x = 30$
* Divide by 2: $x = 15$

3) $\frac{4x + 5}{6} = \frac{7}{2}$
* Cross-multiply to get rid of the fractions: $2(4x + 5) = 6(7)$
* Simplify: $8x + 10 = 42$
* Subtract 10 from both sides: $8x = 32$
* Divide by 8: $x = 4$

4) $8 = 2(x - 5) + 6x$
* Distribute the 2 into the parentheses: $8 = 2x - 10 + 6x$
* Combine like terms ($2x$ and $6x$): $8 = 8x - 10$
* Add 10 to both sides: $18 = 8x$
* Divide by 8: $x = \frac{18}{8}$
* Simplify the fraction: $x = \frac{9}{4}$ (or $2.25$)

5) $-(x + 2) = 2(3x - 4)$
* Distribute the negative sign on the left and the 2 on the right: $-x - 2 = 6x - 8$
* Add $x$ to both sides to move variables to one side: $-2 = 7x - 8$
* Add 8 to both sides: $6 = 7x$
* Divide by 7: $x = \frac{6}{7}$

6) $3 = 4(x - 2) + 5 - 3x$
* Distribute the 4: $3 = 4x - 8 + 5 - 3x$
* Combine like terms on the right side ($4x - 3x$ and $-8 + 5$): $3 = x - 3$
* Add 3 to both sides: $6 = x$
* $x = 6$

7) $\frac{2x - 1}{3} - \frac{3x}{4} = \frac{5}{6}$
* Find the least common multiple for denominators 3, 4, and 6, which is 12. Multiply every term by 12.
* $12(\frac{2x - 1}{3}) - 12(\frac{3x}{4}) = 12(\frac{5}{6})$
* Simplify: $4(2x - 1) - 3(3x) = 2(5)$
* Expand: $8x - 4 - 9x = 10$
* Combine like terms: $-x - 4 = 10$
* Add 4 to both sides: $-x = 14$
* Multiply by -1: $x = -14$

8) $-5x + 3 = 2x + 8$
* Subtract $2x$ from both sides: $-7x + 3 = 8$
* Subtract 3 from both sides: $-7x = 5$
* Divide by -7: $x = -\frac{5}{7}$

9) $4 = -(2x + 4)$
* Distribute the negative sign: $4 = -2x - 4$
* Add 4 to both sides: $8 = -2x$
* Divide by -2: $x = -4$

10) $\frac{7}{8}y - 6 = 8$
* Add 6 to both sides: $\frac{7}{8}y = 14$
* Multiply both sides by the reciprocal $\frac{8}{7}$: $y = 14 \cdot \frac{8}{7}$
* Simplify ($14 \div 7 = 2$): $y = 2 \cdot 8$
* $y = 16$

Final Answer:
1) x = 25
2) x = 15
3) x = 4
4) x = 9/4 (or 2.25)
5) x = 6/7
6) x = 6
7) x = -14
8) x = -5/7
9) x = -4
10) y = 16