Solving linear equations by cross multiplication - Variation Theory - Free Printable

Let's solve each equation one by one. We are given a list of equations, and we need to find the value of $ x $ that satisfies each.

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1. $ \frac{x}{7} = \frac{3}{2} $

Multiply both sides by 7:
$$
x = 7 \cdot \frac{3}{2} = \frac{21}{2}
$$
✔ Solution: $ x = \frac{21}{2} $

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2. $ \frac{14}{2x + 1} = \frac{7}{3} $

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x + 1)
\Rightarrow 42 = 14x + 7
$$
Subtract 7:
$$
35 = 14x \Rightarrow x = \frac{35}{14} = \frac{5}{2}
$$
✔ Solution: $ x = \frac{5}{2} $

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3. $ \frac{2x}{7} = \frac{3}{2} $

Multiply both sides by 7:
$$
2x = \frac{21}{2}
\Rightarrow x = \frac{21}{4}
$$
✔ Solution: $ x = \frac{21}{4} $

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4. $ \frac{14}{2x - 1} = \frac{7}{3} $

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x - 1)
\Rightarrow 42 = 14x - 7
\Rightarrow 49 = 14x \Rightarrow x = \frac{49}{14} = \frac{7}{2}
$$
✔ Solution: $ x = \frac{7}{2} $

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5. $ \frac{2x}{7} = \frac{3}{14} $

Multiply both sides by 7:
$$
2x = \frac{3}{2} \Rightarrow x = \frac{3}{4}
$$
✔ Solution: $ x = \frac{3}{4} $

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6. $ \frac{7}{2x - 1} = \frac{7}{3} $

Since numerators are equal, set denominators equal:
$$
2x - 1 = 3 \Rightarrow 2x = 4 \Rightarrow x = 2
$$
✔ Solution: $ x = 2 $

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7. $ \frac{2x}{14} = \frac{3}{14} $

Multiply both sides by 14:
$$
2x = 3 \Rightarrow x = \frac{3}{2}
$$
✔ Solution: $ x = \frac{3}{2} $

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8. $ \frac{2x - 1}{7} = \frac{7}{3} $

Multiply both sides by 7:
$$
2x - 1 = \frac{49}{3}
\Rightarrow 2x = \frac{49}{3} + 1 = \frac{52}{3}
\Rightarrow x = \frac{52}{6} = \frac{26}{3}
$$
✔ Solution: $ x = \frac{26}{3} $

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9. $ \frac{2x}{14} = \frac{3}{7} $

Simplify left side: $ \frac{x}{7} = \frac{3}{7} $

Multiply both sides by 7:
$$
x = 3
$$
✔ Solution: $ x = 3 $

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10. $ \frac{2x - 1}{6} = \frac{7}{3} $

Multiply both sides by 6:
$$
2x - 1 = 14 \Rightarrow 2x = 15 \Rightarrow x = \frac{15}{2}
$$
✔ Solution: $ x = \frac{15}{2} $

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11. $ \frac{14}{2x} = \frac{7}{3} $

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot 2x \Rightarrow 42 = 14x \Rightarrow x = 3
$$
✔ Solution: $ x = 3 $

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12. $ \frac{1 - 2x}{6} = \frac{7}{3} $

Multiply both sides by 6:
$$
1 - 2x = 14 \Rightarrow -2x = 13 \Rightarrow x = -\frac{13}{2}
$$
✔ Solution: $ x = -\frac{13}{2} $

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✔ Final Answers:

| Equation | Solution |
|--------|---------|
| $ \frac{x}{7} = \frac{3}{2} $ | $ x = \frac{21}{2} $ |
| $ \frac{14}{2x + 1} = \frac{7}{3} $ | $ x = \frac{5}{2} $ |
| $ \frac{2x}{7} = \frac{3}{2} $ | $ x = \frac{21}{4} $ |
| $ \frac{14}{2x - 1} = \frac{7}{3} $ | $ x = \frac{7}{2} $ |
| $ \frac{2x}{7} = \frac{3}{14} $ | $ x = \frac{3}{4} $ |
| $ \frac{7}{2x - 1} = \frac{7}{3} $ | $ x = 2 $ |
| $ \frac{2x}{14} = \frac{3}{14} $ | $ x = \frac{3}{2} $ |
| $ \frac{2x - 1}{7} = \frac{7}{3} $ | $ x = \frac{26}{3} $ |
| $ \frac{2x}{14} = \frac{3}{7} $ | $ x = 3 $ |
| $ \frac{2x - 1}{6} = \frac{7}{3} $ | $ x = \frac{15}{2} $ |
| $ \frac{14}{2x} = \frac{7}{3} $ | $ x = 3 $ |
| $ \frac{1 - 2x}{6} = \frac{7}{3} $ | $ x = -\frac{13}{2} $ |

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Let me know if you'd like these verified or graphed!