Year 10 - Substituting and rearranging formulas - Free practice

Year 10 Substituting into and Rearranging Formulas Worksheets & Practice

Work with real formulas, substitute values, and solve for any variable. Worked solutions every time, free.

Quick answer

How do you substitute into and rearrange formulas?

Substituting means replacing each pronumeral with its given value and evaluating. Rearranging means using inverse operations to isolate a chosen variable. For example, given v = u + at, making a the subject gives a = (v − u) / t. Always check by substituting your answer back into the original formula.

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Worked examples

Year 10 substituting and rearranging formulas: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Substituting into a formula
The formula for the area of a trapezium is $A = \frac{1}{2}(a + b)h$ where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height. Find $A$ when $a = 5$ cm, $b = 11$ cm and $h = 8$ cm.
Show worked solution
Key idea: Replace each variable with its given value, then evaluate using the correct order of operations.
1. Write the formula: $A = \frac{1}{2}(a + b)h$
2. Substitute the values: $A = \frac{1}{2}(5 + 11) \times 8$
3. Simplify inside the brackets: $A = \frac{1}{2} \times 16 \times 8$
4. Evaluate: $A = 64$
Answer: The area of the trapezium is $64$ cm².
Example 2 · Rearranging a formula with a square root
The formula $v = \sqrt{u^2 + 2as}$ relates the final velocity $v$ , initial velocity $u$ , acceleration $a$ and displacement $s$ of an object. Rearrange the formula to make $s$ the subject.
Show worked solution
Key idea: Use inverse operations in reverse order to isolate the desired variable, undoing the outermost operation first.
1. Square both sides to remove the square root: $v^2 = u^2 + 2as$
2. Subtract $u^2$ from both sides: $v^2 - u^2 = 2as$
3. Divide both sides by $2a$ : $s = \frac{v^2 - u^2}{2a}$
Answer: $s = \dfrac{v^2 - u^2}{2a}$
Example 3 · Applying formula rearrangement in context
A scientist uses the formula $E = \frac{mv^2}{2}$ to find the kinetic energy $E$ (in joules) of an object with mass $m$ kg moving at speed $v$ m/s. A ball has a kinetic energy of $360$ J and a mass of $5$ kg. Find the speed of the ball.
Show worked solution
Key idea: Rearrange the formula to make the unknown variable the subject, then substitute the known values.
1. Write the formula and rearrange to make $v$ the subject. Multiply both sides by $2$ : $2E = mv^2$
2. Divide both sides by $m$ : $v^2 = \frac{2E}{m}$
3. Take the positive square root (speed is positive): $v = \sqrt{\frac{2E}{m}}$
4. Substitute $E = 360$ and $m = 5$ : $v = \sqrt{\frac{2 \times 360}{5}} = \sqrt{\frac{720}{5}} = \sqrt{144}$
5. Evaluate: $v = 12$
6. Check by substituting back: $E = \frac{5 \times 12^2}{2} = \frac{5 \times 144}{2} = \frac{720}{2} = 360$ ✓
Answer: The speed of the ball is $12$ m/s.

Common mistakes

Where students usually lose marks

Forgetting to square root at the end

From $v^2 = 144$ , writing the answer as $v = 144$ .

When you reach $v^2 = \text{number}$ , you must take the square root of both sides to find $v$ . Always check what operation was applied to your variable and undo it fully.

Squaring only part of the expression

From $v = \sqrt{u^2 + 2as}$ , squaring to get $v^2 = u + 2as$ (forgetting to square $u$ ).

Squaring removes the square root from the whole expression under the root sign. Every term inside the root is squared together, not individually. In this case $v^2 = u^2 + 2as$ .

Dividing by only part of the denominator

From $2E = mv^2$ , dividing by $m$ to get $v^2 = 2E - m$ instead of $v^2 = \frac{2E}{m}$ .

When you divide both sides by $m$ , every term on the other side is divided by $m$ as a whole. Write it as a fraction: $v^2 = \frac{2E}{m}$ .

Substituting before rearranging

Putting numbers in first and then trying to rearrange an equation full of decimals.

Rearrange the formula with letters first, then substitute numbers into the rearranged version. This keeps working tidy and reduces arithmetic errors.

Try one yourself

Your turn

The formula for the area of a trapezium is $A = \frac{1}{2}(a + b)h$ where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height. A trapezium has an area of $96 \text{ cm}^2$ , one parallel side $a = 10 \text{ cm}$ and height $h = 8 \text{ cm}$ . Find the length of the other parallel side $b$ , in cm.

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About this worksheet

Why we built it

Substituting into and rearranging formulas is a core Year 10 algebra skill that connects to physics, geometry, finance and beyond. These worksheets build from straightforward substitution into one-variable formulas through to multi-step rearrangements where students isolate any chosen variable, with full worked solutions so students can follow every line of reasoning and spot where their own working goes astray.

What's covered

Sub-skills your student will practise

✓Substituting given values into formulas to find an unknown
✓Rearranging a formula to make a specified variable the subject
✓Working with formulas involving squares, square roots and fractions
✓Applying formula rearrangement to science and geometry contexts
✓Checking answers by substituting back into the original formula
✓Handling formulas where the subject appears more than once

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Frequently asked

Questions parents ask about substituting and rearranging formulas

What is the difference between substituting into a formula and rearranging it?

Substituting means replacing variables with known numbers to calculate one remaining unknown. Rearranging means using algebraic steps to rewrite the formula so a different variable is by itself on one side, before any numbers are plugged in.

Why does Year 10 maths use formulas from science and other subjects?

The Australian Curriculum expects students to apply algebra in real contexts, not just abstract exercises. Formulas from physics, geometry and finance give purpose to rearrangement and help students see why the skill matters beyond the maths classroom.

My child can substitute numbers fine but freezes when asked to rearrange. What usually helps?

The most common fix is treating rearrangement like a balance scale: whatever operation undoes a term on one side must be applied to the other. Practising with simple formulas first, then checking by substituting back in, builds that instinct steadily.

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