Multi-Step Equations: Fractions | EdBoost. - Free Printable

Let’s solve each equation one by one. Remember, to solve for the variable, we want to get the variable alone on one side of the equation. Since all these are “one step” equations with fractions, we’ll multiply both sides by the denominator (or reciprocal) to cancel out the fraction.

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1)
Equation:
\(\frac{-6x}{9} = 12\)

Multiply both sides by 9:
\(-6x = 108\)

Divide both sides by -6:
\(x = \frac{108}{-6} = -18\)

✔ Check: \(\frac{-6(-18)}{9} = \frac{108}{9} = 12\) → Correct!

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2)
Equation:
\(\frac{5y}{9} = 60\)

Multiply both sides by 9:
\(5y = 540\)

Divide by 5:
\(y = \frac{540}{5} = 108\)

✔ Check: \(\frac{5(108)}{9} = \frac{540}{9} = 60\) → Correct!

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3)
Equation:
\(\frac{10}{-27} = \frac{2}{3}b\)

We can rewrite as:
\(\frac{2}{3}b = -\frac{10}{27}\)

Multiply both sides by reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\):
\(b = -\frac{10}{27} \cdot \frac{3}{2} = -\frac{30}{54} = -\frac{5}{9}\)

✔ Simplify: Divide numerator and denominator by 6 → \(-\frac{5}{9}\)

Check: \(\frac{2}{3} \cdot (-\frac{5}{9}) = -\frac{10}{27}\) → Correct!

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4)
Equation:
\(\frac{7y}{12} = 84\)

Multiply both sides by 12:
\(7y = 1008\)

Divide by 7:
\(y = \frac{1008}{7} = 144\)

✔ Check: \(\frac{7(144)}{12} = \frac{1008}{12} = 84\) → Correct!

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5)
Equation:
\(\frac{-2z}{12} = 18\)

Simplify left side first: \(\frac{-2z}{12} = \frac{-z}{6}\)
So: \(\frac{-z}{6} = 18\)

Multiply both sides by 6:
\(-z = 108\) → \(z = -108\)

✔ Check: \(\frac{-2(-108)}{12} = \frac{216}{12} = 18\) → Correct!

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6)
Equation:
\(\frac{11w}{14} = -121\)

Multiply both sides by 14:
\(11w = -1694\)

Divide by 11:
\(w = \frac{-1694}{11} = -154\)

✔ Check: \(\frac{11(-154)}{14} = \frac{-1694}{14} = -121\) → Correct!

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7)
Equation:
\(\frac{15y}{-65} = -90\)

Simplify fraction: \(\frac{15}{-65} = -\frac{3}{13}\)
So: \(-\frac{3}{13}y = -90\)

Multiply both sides by \(-\frac{13}{3}\):
\(y = -90 \cdot (-\frac{13}{3}) = 90 \cdot \frac{13}{3} = 30 \cdot 13 = 390\)

✔ Check: \(\frac{15(390)}{-65} = \frac{5850}{-65} = -90\) → Correct!

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8)
Equation:
\(\frac{8y}{14} = \frac{6}{10}\)

Simplify both sides:
Left: \(\frac{8y}{14} = \frac{4y}{7}\)
Right: \(\frac{6}{10} = \frac{3}{5}\)

So: \(\frac{4y}{7} = \frac{3}{5}\)

Cross-multiply:
\(4y \cdot 5 = 3 \cdot 7\) → \(20y = 21\) → \(y = \frac{21}{20}\)

✔ Check: \(\frac{8 \cdot \frac{21}{20}}{14} = \frac{\frac{168}{20}}{14} = \frac{168}{280} = \frac{3}{5} = \frac{6}{10}\) → Correct!

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9)
Equation:
\(\frac{-25y}{100} = 24\)

Simplify: \(\frac{-25y}{100} = -\frac{y}{4}\)
So: \(-\frac{y}{4} = 24\) → Multiply both sides by -4:
\(y = -96\)

✔ Check: \(\frac{-25(-96)}{100} = \frac{2400}{100} = 24\) → Correct!

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10)
Equation:
\(\frac{6}{8} = \frac{3a}{22}\)

Simplify left: \(\frac{6}{8} = \frac{3}{4}\)
So: \(\frac{3}{4} = \frac{3a}{22}\)

Divide both sides by 3:
\(\frac{1}{4} = \frac{a}{22}\) → Multiply both sides by 22:
\(a = \frac{22}{4} = \frac{11}{2} = 5.5\)

✔ Check: \(\frac{3 \cdot 5.5}{22} = \frac{16.5}{22} = 0.75 = \frac{3}{4} = \frac{6}{8}\) → Correct!

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11)
Equation:
\(\frac{3y}{14} = \frac{6}{10}\)

Simplify right: \(\frac{6}{10} = \frac{3}{5}\)
So: \(\frac{3y}{14} = \frac{3}{5}\)

Divide both sides by 3:
\(\frac{y}{14} = \frac{1}{5}\) → Multiply by 14:
\(y = \frac{14}{5} = 2.8\)

✔ Check: \(\frac{3 \cdot 2.8}{14} = \frac{8.4}{14} = 0.6 = \frac{3}{5} = \frac{6}{10}\) → Correct!

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12)
Equation:
\(\frac{9}{-15}g = \frac{-6}{39}\)

Simplify both sides:
Left: \(\frac{9}{-15} = -\frac{3}{5}\) → So: \(-\frac{3}{5}g = -\frac{6}{39}\)
Right: \(\frac{-6}{39} = -\frac{2}{13}\)

So: \(-\frac{3}{5}g = -\frac{2}{13}\)

Multiply both sides by \(-\frac{5}{3}\):
\(g = -\frac{2}{13} \cdot (-\frac{5}{3}) = \frac{10}{39}\)

✔ Check: \(\frac{9}{-15} \cdot \frac{10}{39} = -\frac{3}{5} \cdot \frac{10}{39} = -\frac{30}{195} = -\frac{2}{13} = \frac{-6}{39}\) → Correct!

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Final Answer:

1) x = -18
2) y = 108
3) b = -5/9
4) y = 144
5) z = -108
6) w = -154
7) y = 390
8) y = 21/20
9) y = -96
10) a = 11/2
11) y = 14/5
12) g = 10/39