Finding the equation of a line

Finding the equation of a line from the gradient and a point, from two points, and from a graph, using y = mx + c or y − y₁ = m(x − x₁).

Gradient and a point

Point-gradient form

$$y - y_1 = m(x - x_1)$$

Example 1●●

Find the line with gradient $3$ through $(2, 5)$.

$y - 5 = 3(x - 2)$

$y = 3x - 1$

Example 2●●

Find the line through $(1, 3)$ and $(3, 7)$.

$m = \dfrac{7 - 3}{3 - 1} = 2$

$y - 3 = 2(x - 1)$, so $y = 2x + 1$.

Flashcards

Point-gradient form

$y - y_1 = m(x - x_1)$

Two points

Find $m$ first, then use either point.

Check

Substitute the other point.

Practice

Gradient $3$, through $(2, 5)$: $y = $ ?
Through $(1, 3)$ and $(3, 7)$: $y = $ ?
Through $(-2, 5)$ and $(2, -3)$: $y = $ ?
Gradient $-1$, through $(3, 1)$: $y = $ ?
Gradient $\tfrac{1}{2}$, through $(2, 3)$: $y = $ ?
Through $(0, 0)$ and $(2, 8)$: $y = $ ?
The line through $(1, 2)$ and $(3, k)$ has gradient $1.5$. Find $k$.
Line with $y$-intercept $2$ through $(1, 5)$: $y = $ ?