In short: To rearrange an equation into y = mx + c form, get the y-term on its own on one side, then divide everything by the number in front of y. Once y is alone, the number multiplying x is the gradient m and the constant is the y-intercept c.
Rearranging an equation into y = mx + c form is the skill that unlocks the gradient and y-intercept of any straight line on the GCSE Higher maths paper (AQA). Lines are often given in a jumbled form like 4x + 2y = 10, and you cannot read the gradient until you tidy them up. This guide shows the safe method, a worked example with each step checked, and the slips to avoid.
The reliable method
The goal is to make y the subject — to get y on its own on the left, with everything else on the right. Follow these steps.
- Isolate the y-term. Move every term that does not contain y to the other side of the equals sign, changing each sign as it crosses.
- Divide by the coefficient of y. If y has a number in front of it, divide every term on both sides by that number.
- Tidy the right-hand side. Write it in the order mx + c: the x-term first, then the constant.
- Read off m and c. The number multiplying x is the gradient m; the constant on its own is the y-intercept c.
Do one operation to the whole equation at a time, and apply it to every term — that is what keeps the equation balanced.
A worked example
Rearrange 4x + 2y = 10 into the form y = mx + c and state the gradient and y-intercept.
Step 1 — isolate the y-term. Subtract 4x from both sides:
2y = 10 − 4x
Step 2 — divide by the coefficient of y. The coefficient of y is 2, so divide every term by 2:
y = 5 − 2x
Step 3 — tidy into mx + c order. Write the x-term first:
y = −2x + 5
Step 4 — read off m and c. The gradient is m = −2 and the y-intercept is c = 5.
Check. Substitute x = 1: y = −2 × 1 + 5 = 3. Test in the original: 4 × 1 + 2 × 3 = 4 + 6 = 10. That matches, so the rearrangement is correct.
This works because rearranging never changes which points satisfy the equation — it only rewrites the same line in a form where the gradient and intercept are visible.
Common mistakes to avoid
- Dividing only some terms. When you divide by the coefficient of y, every term must be divided, including the constant. Dividing only the y-term unbalances the equation.
- Sign errors moving terms. A term changes sign as it crosses the equals sign: 4x + 2y = 10 becomes 2y = 10 − 4x, not 2y = 10 + 4x.
- Reading the gradient before rearranging. In 4x + 2y = 10 the gradient is not 4. You must rearrange first; mistaking a coefficient for the gradient is the classic [slope-intercept reversal](/misconceptions/slope-intercept-reversal).
- Leaving the answer as y = 5 − 2x. This is correct but easy to misread. Reorder to y = −2x + 5 so the gradient (−2) is unmistakable.
- Forgetting the sign of the gradient. After dividing −4x by 2 you get −2x, so m is negative. Carry the minus sign through.
Frequently asked questions
How do you rearrange a line equation to find the gradient? Make y the subject: move all non-y terms to the other side, divide every term by the coefficient of y, then write it as y = mx + c. The number multiplying x is the gradient.
What does it mean to make y the subject? It means rearranging the equation so y stands alone on one side, with everything else on the other. Once y is the subject, the equation is in the form y = mx + c.
Why do I have to divide every term, not just the y-term? To keep the equation balanced. Whatever you do to one side you must do to the entire side, so every term is divided by the same number. Dividing only one term changes the equation's meaning.
Where do I find the gradient and y-intercept once it is rearranged? In y = mx + c, the coefficient of x (the number directly in front of x) is the gradient m, and the constant term on its own is the y-intercept c.
Can the gradient be a fraction after rearranging? Yes. Dividing by the coefficient of y often produces a fractional gradient, such as y = (3/2)x + 1. A fractional gradient is perfectly valid and common in exam questions.
Practise this
See which slips cost you marks — [take the free diagnostic](/diagnostic). Related: [slope-intercept reversal explained](/misconceptions/slope-intercept-reversal).