Worksheet for solving multi-step equations involving fractions.

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

1) $5(2b + \frac{1}{3}) = 7b + 11$

* Step 1: Distribute the 5 into the parentheses.
$10b + \frac{5}{3} = 7b + 11$
* Step 2: Subtract $7b$ from both sides to get the variables on one side.
$3b + \frac{5}{3} = 11$
* Step 3: Subtract $\frac{5}{3}$ from both sides. To do this, change 11 into a fraction with a denominator of 3 ($\frac{33}{3}$).
$3b = \frac{33}{3} - \frac{5}{3}$
$3b = \frac{28}{3}$
* Step 4: Divide by 3 (which is the same as multiplying by $\frac{1}{3}$).
$b = \frac{28}{3} \cdot \frac{1}{3}$
$b = \frac{28}{9}$

2) $\frac{9}{7}(1 + x) - \frac{6}{7}x + 1 = 1$

* Step 1: Subtract 1 from both sides.
$\frac{9}{7}(1 + x) - \frac{6}{7}x = 0$
* Step 2: Distribute the $\frac{9}{7}$.
$\frac{9}{7} + \frac{9}{7}x - \frac{6}{7}x = 0$
* Step 3: Combine the $x$ terms ($\frac{9}{7} - \frac{6}{7} = \frac{3}{7}$).
$\frac{9}{7} + \frac{3}{7}x = 0$
* Step 4: Subtract $\frac{9}{7}$ from both sides.
$\frac{3}{7}x = -\frac{9}{7}$
* Step 5: Multiply both sides by $\frac{7}{3}$ to isolate $x$.
$x = -\frac{9}{7} \cdot \frac{7}{3}$
$x = -\frac{9}{3}$
$x = -3$

3) $\frac{7}{8}t - \frac{4}{7} = \frac{5}{6}t - \frac{3}{4}$

* Step 1: Find the Least Common Denominator (LCD) for 8, 7, 6, and 4, which is 168. Multiply every term by 168 to clear the fractions.
$168(\frac{7}{8}t) - 168(\frac{4}{7}) = 168(\frac{5}{6}t) - 168(\frac{3}{4})$
$21(7t) - 24(4) = 28(5t) - 42(3)$
$147t - 96 = 140t - 126$
* Step 2: Subtract $140t$ from both sides.
$7t - 96 = -126$
* Step 3: Add 96 to both sides.
$7t = -30$
* Step 4: Divide by 7.
$t = -\frac{30}{7}$

4) $\frac{8}{5} + 3z = 10z - 4$

* Step 1: Subtract $3z$ from both sides.
$\frac{8}{5} = 7z - 4$
* Step 2: Add 4 to both sides. Convert 4 to $\frac{20}{5}$.
$\frac{8}{5} + \frac{20}{5} = 7z$
$\frac{28}{5} = 7z$
* Step 3: Divide by 7 (multiply by $\frac{1}{7}$).
$z = \frac{28}{5} \cdot \frac{1}{7}$
$z = \frac{4}{5}$

5) $\frac{3}{2}h - 10 = 4h$

* Step 1: Subtract $\frac{3}{2}h$ from both sides. Note that $4h$ is the same as $\frac{8}{2}h$.
$-10 = \frac{8}{2}h - \frac{3}{2}h$
$-10 = \frac{5}{2}h$
* Step 2: Multiply by the reciprocal $\frac{2}{5}$.
$h = -10 \cdot \frac{2}{5}$
$h = -\frac{20}{5}$
$h = -4$

6) $\frac{3 + 4n}{4} = \frac{8}{4}n$

* Step 1: Simplify the right side. $\frac{8}{4}n$ becomes $2n$.
$\frac{3 + 4n}{4} = 2n$
* Step 2: Multiply both sides by 4 to remove the denominator.
$3 + 4n = 8n$
* Step 3: Subtract $4n$ from both sides.
$3 = 4n$
* Step 4: Divide by 4.
$n = \frac{3}{4}$

7) $\frac{u - 5}{8} = 2u - 1$

* Step 1: Multiply both sides by 8.
$u - 5 = 8(2u - 1)$
* Step 2: Distribute the 8.
$u - 5 = 16u - 8$
* Step 3: Subtract $u$ from both sides.
$-5 = 15u - 8$
* Step 4: Add 8 to both sides.
$3 = 15u$
* Step 5: Divide by 15.
$u = \frac{3}{15}$
* Step 6: Simplify the fraction.
$u = \frac{1}{5}$

8) $6v + \frac{4}{9} = 4(\frac{1}{2} + v)$

* Step 1: Distribute the 4 on the right side.
$6v + \frac{4}{9} = 2 + 4v$
* Step 2: Subtract $4v$ from both sides.
$2v + \frac{4}{9} = 2$
* Step 3: Subtract $\frac{4}{9}$ from both sides. Convert 2 to $\frac{18}{9}$.
$2v = \frac{18}{9} - \frac{4}{9}$
$2v = \frac{14}{9}$
* Step 4: Divide by 2 (multiply by $\frac{1}{2}$).
$v = \frac{14}{9} \cdot \frac{1}{2}$
$v = \frac{7}{9}$

Final Answer:
1) $b = \frac{28}{9}$
2) $x = -3$
3) $t = -\frac{30}{7}$
4) $z = \frac{4}{5}$
5) $h = -4$
6) $n = \frac{3}{4}$
7) $u = \frac{1}{5}$
8) $v = \frac{7}{9}$