Equation in the Form ax + by + c = 0

Move a straight-line equation between y = mx + c and ax + by + c = 0, and read the gradient from either form.

In the exam for Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 3 real past-paper questions below

One Line, Two Ways to Write It

The same line can be written two ways. They give exactly the same points.

Gradient form

y = 2 x - 5

shows m and c

General form

2 x - y - 5 = 0

everything on one side

Rule

For $a x + b y + c = 0$ : every term on one side, 0 on the other, and a, b and c whole numbers.

From y = mx + c to ax + by + c = 0

Questions often say “give your answer in the form $a x + b y + c = 0$ ”. Clear any fraction first, then move everything to one side.

Worked example

1
$y = \frac{2}{3} x + 4$ — The line. The $\frac{2}{3}$ is a fraction, and the form needs whole numbers, so clear it first.
2
$3 y = 2 x + 12$ — Times every term by 3, the bottom of the fraction: $\frac{2}{3} x \times 3 = 2 x$ and $4 \times 3 = 12$ .
3
$0 = 2 x - 3 y + 12$ — Take 3y from both sides, so one side is 0. Moving the y term keeps the x term positive.
4
$2 x - 3 y + 12 = 0$ — Swap the sides. Same equation, now in the order ax + by + c = 0, with a = 2, b = −3, c = 12.

Common mistake

Multiplying only the fraction: $3 y = 2 x + 4$ . Every term gets multiplied, so the 4 becomes 12.

Gradient From ax + by + c = 0

You cannot read the gradient straight from $a x + b y + c = 0$ . Get y on its own first.

Worked example

1
$3 x + 4 y - 8 = 0$ — The line. y is not on its own, so the gradient is hidden.
2
$4 y - 8 = - 3 x$ — Take 3x from both sides, to clear the x term from the left.
3
$4 y = - 3 x + 8$ — Add 8 to both sides. Now only the y term is on the left.
4
$y = - \frac{3}{4} x + 2$ — Divide every term by 4, so y stands alone: $8 \div 4 = 2$ .
5
$gradient = - \frac{3}{4}, y-intercept = 2$ — Read them off. The gradient is the number in front of x; the y-intercept is the number on its own.

Common mistake

Saying the gradient is 3. The 3 is only the gradient once y is on its own with nothing in front.

When the Equation Looks Different

Exams often write the line another way, like “3y + 2x = 15”: the y term first and the number on the right. The method does not change: get y on its own.

Worked example

1
$3 y + 2 x = 15$ — The line. y is not alone, so the gradient is hidden.
2
$3 y = 15 - 2 x$ — Take 2x from both sides, so only the y term is on the left.
3
$3 y = - 2 x + 15$ — Reorder the right side so the x term comes first, like y = mx + c. The minus stays with the 2x.
4
$y = - \frac{2}{3} x + 5$ — Divide every term by 3, so y stands alone: $15 \div 3 = 5$ .
5
$gradient = - \frac{2}{3}, y-intercept = 5$ — Read them off: the gradient is in front of x, the y-intercept is the number on its own.

Key idea

Write the $3 y = 15 - 2 x$ line before the answer. It earns a mark for working even if your final gradient is wrong.

Common mistake

Dividing only the 15 by 3 and writing $y = - 2 x + 5$ . Every term is divided, so the gradient is $- \frac{2}{3}$ , not −2.

Shortcut: Gradient = −a ÷ b

Textbooks print a shortcut that reads the gradient straight from ax + by + c = 0. Use it to check your rearranging.

Formula

gradient = - \frac{a}{b} y-intercept = - \frac{c}{b}

$a$ is the number in front of x, $b$ the number in front of y, and $c$ the number on its own. Each one keeps its sign.

Worked example

1
$5 x - 2 y + 8 = 0$ — The line. Everything is on one side with 0 on the other, so the shortcut works.
2
$a = 5, b = - 2, c = 8$ — Read the numbers with their signs. The y term is −2y, so b is −2, not 2.
3
$gradient = - \frac{5}{( - 2 )}$ — Put a and b in the formula. Keep −2 in brackets so its sign is not lost.
4
$= \frac{5}{2}$ — The minus in front and the minus of −2 cancel: minus divided by minus is plus.
5
$y-intercept = - \frac{8}{( - 2 )}$ — Same formula idea for c: put c and b in.
6
$= 4$ — Minus divided by minus is plus again, and 8 ÷ 2 = 4. Check: y = 2.5x + 4 gives the same line.

Common mistake

Using the shortcut on 3y + 2x = 15. It only works with 0 on one side, so first write $2 x + 3 y - 15 = 0$ .

Which Method? Read the Question

Which way are you going?

You see	You do
“Give it as ax + by + c = 0”	clear fractions, all to one side
Gradient of ax + by + c = 0	get y on its own
Two points, answer as ax + by + c = 0	Equation of a Line Through Two Points
Reading m and c	The Equation y = mx + c

Key idea

Quick check: for $a x + b y + c = 0$ the gradient is $- \frac{a}{b}$ . For $3 x + 4 y - 8 = 0$ that is $- \frac{3}{4}$ .

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