Equations of lines

Horizontal and vertical lines, gradient–intercept form y = mx + b, and general form ax + by + c = 0, and converting between them.

Horizontal and vertical lines

Every point on a horizontal line has the same $y$-coordinate, so its equation is $y = k$. Every point on a vertical line has the same $x$-coordinate: $x = h$. The $x$-axis is $y = 0$ and the $y$-axis is $x = 0$.

Two standard forms

Gradient–intercept form

$$y = mx + b$$

Gradient $m$, $y$-intercept $b$.

General form

$$ax + by + c = 0$$

Integer coefficients, usually with $a > 0$. Every line, including vertical ones, can be written this way.

General form explorer

Change a, b and c in ax + by + c = 0 and read the gradient and intercepts.

Try this

Which coefficient must be zero for a horizontal line? For a vertical line?

Example 1●

Write $y = -\tfrac23 x + 4$ in general form.

$3y = -2x + 12$

Multiply by 3 to clear the fraction.

$2x + 3y - 12 = 0$

Example 2●●

Find the gradient and both intercepts of $4x - 5y + 20 = 0$.

$5y = 4x + 20$, so $y = \tfrac45 x + 4$: gradient $\tfrac45$, $y$-intercept $4$.

$y = 0$: $4x = -20$, so the $x$-intercept is $-5$.

Flashcards

Horizontal line through (2, 5)

$y = 5$

Vertical line through (2, 5)

$x = 2$

Gradient of ax + by + c = 0

$-\dfrac{a}{b}$

General form

$ax + by + c = 0$, integers, $a > 0$.

Practice

The vertical line through $(-3, 4)$ is:
Find the gradient of $3x + 2y - 7 = 0$.
Find the $y$-intercept of $x - 2y + 8 = 0$.
$y = \tfrac12 x - 3$ in general form is:
Which line is parallel to $y = 3x - 1$?
Find the $x$-intercept of $2x - 3y + 12 = 0$.
The $x$-axis has equation:
For which $k$ does $kx + y - 3 = 0$ have gradient $-5$?