Multi-Step Equations with Fractions Worksheets - Free Printable

Let's solve each of these multi-step equations involving fractions step by step. I'll go through each one carefully, showing the algebraic steps and simplifications.

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1) $ 5\left(2h + \frac{1}{3}\right) = 7h + 11 $

Step 1: Distribute the 5 on the left side:
$$
5 \cdot 2h + 5 \cdot \frac{1}{3} = 7h + 11
\Rightarrow 10h + \frac{5}{3} = 7h + 11
$$

Step 2: Subtract $7h$ from both sides:
$$
10h - 7h + \frac{5}{3} = 11 \Rightarrow 3h + \frac{5}{3} = 11
$$

Step 3: Subtract $\frac{5}{3}$ from both sides:
$$
3h = 11 - \frac{5}{3} = \frac{33}{3} - \frac{5}{3} = \frac{28}{3}
$$

Step 4: Divide both sides by 3:
$$
h = \frac{28}{3} \div 3 = \frac{28}{9}
$$

✔ Answer: $ h = \boxed{\frac{28}{9}} $

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

Step 1: Distribute $ \frac{9}{7} $:
$$
\frac{9}{7} + \frac{9}{7}x - \frac{6}{7}x + 1 = 1
$$

Step 2: Combine like terms:
- $ \frac{9}{7}x - \frac{6}{7}x = \frac{3}{7}x $
- Constants: $ \frac{9}{7} + 1 = \frac{9}{7} + \frac{7}{7} = \frac{16}{7} $

So:
$$
\frac{3}{7}x + \frac{16}{7} = 1
$$

Step 3: Subtract $ \frac{16}{7} $ from both sides:
$$
\frac{3}{7}x = 1 - \frac{16}{7} = \frac{7}{7} - \frac{16}{7} = -\frac{9}{7}
$$

Step 4: Multiply both sides by $ \frac{7}{3} $:
$$
x = -\frac{9}{7} \cdot \frac{7}{3} = -\frac{9}{3} = -3
$$

✔ Answer: $ x = \boxed{-3} $

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

Step 1: Get all terms with $ t $ on one side. Subtract $ \frac{5}{6}t $ from both sides:
$$
\frac{7}{8}t - \frac{4}{7}t - \frac{5}{6}t = -\frac{3}{4}
$$

Step 2: Find a common denominator for the coefficients (LCM of 8, 7, 6 is 168):
Convert each fraction:
- $ \frac{7}{8} = \frac{147}{168} $
- $ \frac{4}{7} = \frac{96}{168} $
- $ \frac{5}{6} = \frac{140}{168} $

Now:
$$
\left(\frac{147}{168} - \frac{96}{168} - \frac{140}{168}\right)t = -\frac{3}{4}
\Rightarrow \left(\frac{147 - 96 - 140}{168}\right)t = -\frac{3}{4}
\Rightarrow \frac{-89}{168}t = -\frac{3}{4}
$$

Step 3: Solve for $ t $:
$$
t = -\frac{3}{4} \div \left(-\frac{89}{168}\right) = \frac{3}{4} \cdot \frac{168}{89}
= \frac{3 \cdot 168}{4 \cdot 89} = \frac{504}{356}
$$

Simplify:
- Divide numerator and denominator by 4: $ \frac{126}{89} $

✔ Answer: $ t = \boxed{\frac{126}{89}} $

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

Step 1: Move all $ z $-terms to one side and constants to the other.
Subtract $ 3z $ from both sides:
$$
\frac{8}{5} = 7z - 4
$$

Add 4 to both sides:
$$
\frac{8}{5} + 4 = 7z \Rightarrow \frac{8}{5} + \frac{20}{5} = \frac{28}{5} = 7z
$$

Step 2: Divide both sides by 7:
$$
z = \frac{28}{5} \div 7 = \frac{28}{5} \cdot \frac{1}{7} = \frac{4}{5}
$$

✔ Answer: $ z = \boxed{\frac{4}{5}} $

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

Step 1: Subtract $ \frac{3}{2}h $ from both sides:
$$
-10 = 4h - \frac{3}{2}h
$$

Convert $ 4h = \frac{8}{2}h $, so:
$$
\frac{8}{2}h - \frac{3}{2}h = \frac{5}{2}h
\Rightarrow -10 = \frac{5}{2}h
$$

Step 2: Multiply both sides by $ \frac{2}{5} $:
$$
h = -10 \cdot \frac{2}{5} = -\frac{20}{5} = -4
$$

✔ Answer: $ h = \boxed{-4} $

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

Step 1: Eliminate denominators by multiplying both sides by $ 4n $ (LCD):
$$
4n \cdot \frac{3 + 4n}{4} = 4n \cdot \frac{8}{n}
\Rightarrow n(3 + 4n) = 32
$$

Step 2: Expand:
$$
3n + 4n^2 = 32
\Rightarrow 4n^2 + 3n - 32 = 0
$$

Step 3: Solve quadratic equation using quadratic formula:
$$
n = \frac{-3 \pm \sqrt{3^2 - 4(4)(-32)}}{2(4)} = \frac{-3 \pm \sqrt{9 + 512}}{8} = \frac{-3 \pm \sqrt{521}}{8}
$$

Since $ \sqrt{521} $ is not a perfect square, leave as is.

✔ Answer: $ n = \boxed{\frac{-3 \pm \sqrt{521}}{8}} $

> Note: This is an irrational solution; check if it was intended to be rational. But based on the equation, this is correct.

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

Step 1: Multiply both sides by 8 to eliminate denominator:
$$
u - 5 = 8(2u - 1) = 16u - 8
$$

Step 2: Subtract $ u $ from both sides:
$$
-5 = 15u - 8
$$

Add 8 to both sides:
$$
3 = 15u \Rightarrow u = \frac{3}{15} = \frac{1}{5}
$$

✔ Answer: $ u = \boxed{\frac{1}{5}} $

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

Step 1: Distribute on the right:
$$
6v + \frac{4}{9} = 4 \cdot \frac{1}{2} + 4v = 2 + 4v
$$

Step 2: Subtract $ 4v $ from both sides:
$$
2v + \frac{4}{9} = 2
$$

Step 3: Subtract $ \frac{4}{9} $:
$$
2v = 2 - \frac{4}{9} = \frac{18}{9} - \frac{4}{9} = \frac{14}{9}
$$

Step 4: Divide by 2:
$$
v = \frac{14}{9} \div 2 = \frac{14}{18} = \frac{7}{9}
$$

✔ Answer: $ v = \boxed{\frac{7}{9}} $

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✔ Final Answers:

1) $ h = \boxed{\frac{28}{9}} $
2) $ x = \boxed{-3} $
3) $ t = \boxed{\frac{126}{89}} $
4) $ z = \boxed{\frac{4}{5}} $
5) $ h = \boxed{-4} $
6) $ n = \boxed{\frac{-3 \pm \sqrt{521}}{8}} $
7) $ u = \boxed{\frac{1}{5}} $
8) $ v = \boxed{\frac{7}{9}} $

Let me know if you'd like these checked or explained further!