Writing Equations of Lines

By the end of this lesson, you will be able to write the equation of a line from whatever you are given, choose the form that makes the job shortest, and check your answer against the information you started with.

Every Question Is the Same Question

A line needs exactly two facts: a direction and a position. Slope and one point. Everything below is a different way of handing you those two.

Given	What is missing	Route
Slope and $y$-intercept	nothing	Write $y = mx + b$ directly
Slope and a point	the intercept	Point-slope, then tidy
Two points	the slope	Work out $m$, then point-slope
A graph	both, but readable	Read $b$ off the axis, count the staircase
A table	the slope	Take differences, then use a row as the point
A situation	both, in words	Rate is the slope, starting amount is $b$

Recognizing which row you are in takes a second and decides the whole answer. Doing the wrong route is not wrong, just slower.

Example 1: Slope and a Point

Here is the whole method on one, worked all the way through. Write the line with slope $5$ through $(2, 3)$.

• You have a slope and a point. Substitute into point-slope form y - 3 = 5(x - 2)
• Multiply out y - 3 = 5x - 10
• Make y the subject y = 5x - 7
• Check it. What does the equation give when x is 2? 3

Worked Example 1: Two Points

Write the line through $(-1, 2)$ and $(3, 14)$.

From a Table

A table gives the slope through differences and a point in any row.

$x$	$1$	$3$	$5$	$7$
$y$	$9$	$5$	$1$	$-3$

$x$ steps by $2$, $y$ steps by $-4$, so $m = \frac{-4}{2} = -2$.

Take the row $(1, 9)$: $y - 9 = -2(x - 1)$, so $y = -2x + 11$.

The check is free here because the table has spare rows. At $x = 5$ the equation gives $-10 + 11 = 1$, which is what the table says. A table is the easiest case to be confident about, so it is worth using the spare rows rather than trusting the arithmetic.

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