Rearranging Formulae (A) | Cazoom Maths Worksheets - Free Printable

Let's solve each of these problems step-by-step, rearranging the given formulae to express $ x $ in terms of $ y $.

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1) $ y = x + 5 $

Subtract 5 from both sides:
$$
x = y - 5
$$

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2) $ y = x - 4 $

Add 4 to both sides:
$$
x = y + 4
$$

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3) $ y = -6 - x $

Add $ x $ to both sides:
$$
y + x = -6 \Rightarrow x = -6 - y
$$

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4) $ y = 9x $

Divide both sides by 9:
$$
x = \frac{y}{9}
$$

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5) $ y = -3x $

Divide both sides by -3:
$$
x = -\frac{y}{3}
$$

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6) $ y = \frac{x}{7} $

Multiply both sides by 7:
$$
x = 7y
$$

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7) $ y = -\frac{2}{x} $

Multiply both sides by $ x $:
$$
yx = -2 \Rightarrow x = -\frac{2}{y}
$$

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8) $ y = 2x + 5 $

Subtract 5:
$$
y - 5 = 2x \Rightarrow x = \frac{y - 5}{2}
$$

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9) $ y = 8x - 7 $

Add 7:
$$
y + 7 = 8x \Rightarrow x = \frac{y + 7}{8}
$$

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10) $ y = 10 - 3x $

Subtract 10:
$$
y - 10 = -3x \Rightarrow x = \frac{10 - y}{3}
$$

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11) $ y = 7(3x + 2) $

Divide both sides by 7:
$$
\frac{y}{7} = 3x + 2
$$
Subtract 2:
$$
\frac{y}{7} - 2 = 3x \Rightarrow x = \frac{1}{3}\left(\frac{y}{7} - 2\right) = \frac{y - 14}{21}
$$

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12) $ y = -4(x + 11) $

Divide both sides by -4:
$$
\frac{y}{-4} = x + 11 \Rightarrow x = -\frac{y}{4} - 11
$$

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13) $ y = \frac{x}{4} + 5 $

Subtract 5:
$$
y - 5 = \frac{x}{4} \Rightarrow x = 4(y - 5) = 4y - 20
$$

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14) $ y = 10 - \frac{x}{3} $

Subtract 10:
$$
y - 10 = -\frac{x}{3} \Rightarrow x = -3(y - 10) = -3y + 30
$$

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15) $ y = \frac{x + 7}{5} $

Multiply both sides by 5:
$$
5y = x + 7 \Rightarrow x = 5y - 7
$$

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16) $ y = \frac{7}{x - 4} $

Multiply both sides by $ x - 4 $:
$$
y(x - 4) = 7 \Rightarrow x - 4 = \frac{7}{y} \Rightarrow x = \frac{7}{y} + 4
$$

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17) $ y = \frac{-8}{9 - x} $

Multiply both sides by $ 9 - x $:
$$
y(9 - x) = -8 \Rightarrow 9y - yx = -8
\Rightarrow -yx = -8 - 9y \Rightarrow x = \frac{8 + 9y}{y}
$$

Alternatively:
$$
x = 9 + \frac{8}{y}
$$
Check: From $ y = \frac{-8}{9 - x} $, solve:
$$
9 - x = \frac{-8}{y} \Rightarrow x = 9 + \frac{8}{y}
$$

✔ So:
$$
x = 9 + \frac{8}{y}
$$

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18) $ y = x^2 $

Take square root:
$$
x = \pm\sqrt{y}
$$

(Note: Two solutions due to squaring)

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19) $ y = 5x^2 $

Divide by 5:
$$
x^2 = \frac{y}{5} \Rightarrow x = \pm\sqrt{\frac{y}{5}}
$$

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20) $ y = 81x^2 $

$$
x^2 = \frac{y}{81} \Rightarrow x = \pm\frac{\sqrt{y}}{9}
$$

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21) $ y = \frac{x^2}{3} $

Multiply by 3:
$$
3y = x^2 \Rightarrow x = \pm\sqrt{3y}
$$

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22) $ y = x^2 + 7 $

Subtract 7:
$$
x^2 = y - 7 \Rightarrow x = \pm\sqrt{y - 7}
$$

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23) $ y = 67 - x^2 $

Rearrange:
$$
x^2 = 67 - y \Rightarrow x = \pm\sqrt{67 - y}
$$

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24) $ y = \frac{5x^2 - 7}{13} $

Multiply both sides by 13:
$$
13y = 5x^2 - 7 \Rightarrow 5x^2 = 13y + 7 \Rightarrow x^2 = \frac{13y + 7}{5}
\Rightarrow x = \pm\sqrt{\frac{13y + 7}{5}}
$$

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25) $ y = \sqrt{11x} $

Square both sides:
$$
y^2 = 11x \Rightarrow x = \frac{y^2}{11}
$$

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26) $ y = \sqrt{x + 2} $

Square both sides:
$$
y^2 = x + 2 \Rightarrow x = y^2 - 2
$$

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27) $ y = \sqrt{10 - x} $

Square both sides:
$$
y^2 = 10 - x \Rightarrow x = 10 - y^2
$$

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28) $ y = \sqrt{\frac{8x - 3}{2}} $

Square both sides:
$$
y^2 = \frac{8x - 3}{2} \Rightarrow 2y^2 = 8x - 3 \Rightarrow 8x = 2y^2 + 3 \Rightarrow x = \frac{2y^2 + 3}{8}
$$

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29) $ y = 5 + \sqrt{\frac{6}{2x + 3}} $

Subtract 5:
$$
y - 5 = \sqrt{\frac{6}{2x + 3}}
$$
Square both sides:
$$
(y - 5)^2 = \frac{6}{2x + 3}
\Rightarrow 2x + 3 = \frac{6}{(y - 5)^2}
\Rightarrow 2x = \frac{6}{(y - 5)^2} - 3
\Rightarrow x = \frac{1}{2}\left( \frac{6}{(y - 5)^2} - 3 \right)
$$

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30) $ y = \frac{8 - x}{4} $

Multiply both sides by 4:
$$
4y = 8 - x \Rightarrow x = 8 - 4y
$$

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## ✔ Final Answers:

| Question | Answer |
|--------|--------|
| 1 | $ x = y - 5 $ |
| 2 | $ x = y + 4 $ |
| 3 | $ x = -6 - y $ |
| 4 | $ x = \frac{y}{9} $ |
| 5 | $ x = -\frac{y}{3} $ |
| 6 | $ x = 7y $ |
| 7 | $ x = -\frac{2}{y} $ |
| 8 | $ x = \frac{y - 5}{2} $ |
| 9 | $ x = \frac{y + 7}{8} $ |
| 10 | $ x = \frac{10 - y}{3} $ |
| 11 | $ x = \frac{y - 14}{21} $ |
| 12 | $ x = -\frac{y}{4} - 11 $ |
| 13 | $ x = 4y - 20 $ |
| 14 | $ x = -3y + 30 $ |
| 15 | $ x = 5y - 7 $ |
| 16 | $ x = \frac{7}{y} + 4 $ |
| 17 | $ x = 9 + \frac{8}{y} $ |
| 18 | $ x = \pm\sqrt{y} $ |
| 19 | $ x = \pm\sqrt{\frac{y}{5}} $ |
| 20 | $ x = \pm\frac{\sqrt{y}}{9} $ |
| 21 | $ x = \pm\sqrt{3y} $ |
| 22 | $ x = \pm\sqrt{y - 7} $ |
| 23 | $ x = \pm\sqrt{67 - y} $ |
| 24 | $ x = \pm\sqrt{\frac{13y + 7}{5}} $ |
| 25 | $ x = \frac{y^2}{11} $ |
| 26 | $ x = y^2 - 2 $ |
| 27 | $ x = 10 - y^2 $ |
| 28 | $ x = \frac{2y^2 + 3}{8} $ |
| 29 | $ x = \frac{1}{2}\left( \frac{6}{(y - 5)^2} - 3 \right) $ |
| 30 | $ x = 8 - 4y $ |

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🔍 Notes:

- For equations involving squares or square roots, remember that $ x = \pm \text{something} $ unless context restricts to positive values.
- Always check for domain restrictions (e.g., denominators ≠ 0, expressions under square roots ≥ 0).
- When solving, isolate the variable step by step using inverse operations.

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