Algebra

Rearranging formulae

Make a different quantity the subject.

Predict, test and explain

Step 1 of 3

Predict

If y = 3x + 2 and y = 14, what is x?

y=ax+b

Drag a highlighted card across =, or tap it. Arrow keys also move cards.

Left side

y

Right side

ax

Move the highlighted term to the other side.

Current values

x=4

Equivalent equation

14=3x+2

Move +b first, then the nonzero factor a. Division applies to the whole expression y − b.

Check the working

Operation	Formula	Using current values
Start	$y=ax+b$	$14=3x+2$

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Make the input the subject of this formula.

$y=3x-4\qquad\text{(make }x\text{ the subject)}$

Hint

Undo subtraction first, then undo multiplication.

Reveal answer

Add four to both sides, then divide the whole resulting side by three.

$x=\frac{y+4}{3}$

Try a short practice

Use these fixed values. The diagram controls are separate.

Check your understanding

Make x the subject.

$y=4x+6$

Complete the missing step

Complete the final step to make h the subject. The base is positive.

$A=\frac{bh}{2},\qquad b>0$

$2A=bh$

Spot the first mistake

This is an attempted solution containing a mistake.

Which step contains the first mistake?

$y=3x-6$

$y+6=3x$
$y+2=x$
$x=y+2$

Build an explanation

Hint 1

Find the desired subject and undo its surrounding operations in reverse order, acting on both whole sides.

Hint 2

For y = ax + b, subtract the complete term b from both sides, then divide by a if a ≠ 0. Keep y − b together in brackets.

Hint 3

For A = bh/2, multiply both sides by 2 to undo the divisor, then divide by b if b ≠ 0. Additive terms, factors and divisors have different inverse operations.

Worked example

This example uses fixed values, separate from the diagram controls.

Make x the subject of y = ax + b with a ≠ 0, and h the subject of A = bh/2 with b ≠ 0.

For y = ax + b, subtract b from both sides: y − b = ax.
Divide both sides by the non-zero factor a: x = (y − b)/a.
For A = bh/2, multiply both sides by 2: 2A = bh.
Divide both sides by the non-zero factor b: h = 2A/b.

Answer: x = (y − b)/a for a ≠ 0; h = 2A/b for b ≠ 0.

Connect this idea

Watch out: A factor does not become an additive term when it changes sides. Dividing both sides means dividing the entire side; (y − b)/a is not y − b/a. A zero coefficient cannot be undone by division.