Math Core

Slope-intercept form is perfect when you know where a line crosses the $y$ -axis. But often you know some other point on the line, like "at 3 hours the candle was 18 cm tall." Point-slope form lets you write the equation directly from any point and the slope, with no need to hunt for $b$ first.

Where the form comes from

Suppose a line has slope $m$ and passes through a known point $(x_1, y_1)$ . Let $(x, y)$ stand for any other point on the line. The slope between those two points must be $m$ :

\frac{y - y_1}{x - x_1} = m.

Multiply both sides by $x - x_1$ to clear the fraction, and you get point-slope form.

Definition

Point-slope form

The point-slope form of the line with slope $m$ through the point $(x_1, y_1)$ is

y - y_1 = m(x - x_1).

Here $x_1$ , $y_1$ and $m$ are specific numbers, while $x$ and $y$ stay as variables. So the equation describes every point $(x, y)$ on the line.

Worked example: From a point and a slope

Write an equation of the line through $(3, -2)$ with slope $4$ . Then rewrite it in slope-intercept form.

Substitute $x_1 = 3$ , $y_1 = -2$ and $m = 4$ :

y - (-2) = 4(x - 3), \quad\text{which simplifies to}\quad y + 2 = 4(x - 3).

That is already a correct answer in point-slope form. To get slope-intercept form, distribute and solve for $y$ :

\begin{aligned} y + 2 &= 4x - 12 && \text{distribute} \\ y &= 4x - 14 && \text{subtract } 2 \end{aligned}

Check: when $x = 3$ , $y = 12 - 14 = -2$ , so the line passes through $(3, -2)$ as required.

Reading the point and the slope

Point-slope form always has minus signs in front of $y_1$ and $x_1$ . That means the numbers you see in the equation have the opposite signs of the coordinates.

In $y - 5 = 2(x - 1)$ , the point is $(1, 5)$ and the slope is $2$ .
In $y + 5 = -2(x - 1)$ , rewrite it as $y - (-5) = -2(x - 1)$ . The point is $(1, -5)$ and the slope is $-2$ .
In $y - 4 = \dfrac{1}{3}(x + 6)$ , rewrite $x + 6$ as $x - (-6)$ . The point is $(-6, 4)$ and the slope is $\dfrac{1}{3}$ .

Common mistake

The most common mistake is keeping the signs as written. In $y + 5 = -2(x - 1)$ , the point is not $(-1, 5)$ . Ask: "What value of $x$ makes $x - 1$ zero? What value of $y$ makes $y + 5$ zero?" Those values, $x = 1$ and $y = -5$ , give the point $(1, -5)$ .

Lines through two points

Point-slope form makes the two-point problem quick: find the slope, then plug in either point.

Equation of a line through two points

Find the slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ .
Substitute $m$ and either point into $y - y_1 = m(x - x_1)$ .
If the question asks for slope-intercept form, distribute and solve for $y$ .

Worked example: From two points

Write the equation of the line through $(-2, 7)$ and $(4, -2)$ in slope-intercept form.

Slope. $m = \dfrac{-2 - 7}{4 - (-2)} = \dfrac{-9}{6} = -\dfrac{3}{2}$ .

Point-slope. Using $(-2, 7)$ : $\;y - 7 = -\dfrac{3}{2}(x + 2)$ .

Slope-intercept.

\begin{aligned} y - 7 &= -\frac{3}{2}x - 3 && \text{distribute: } -\tfrac{3}{2} \cdot 2 = -3 \\ y &= -\frac{3}{2}x + 4 && \text{add } 7 \end{aligned}

The line through (−2, 7) and (4, −2) has slope −3/2 and y-intercept 4.Open in grapher →

Tip

If you had used the other point, $(4, -2)$ , you'd get $y + 2 = -\dfrac{3}{2}(x - 4)$ . It looks different, but distributing gives $y + 2 = -\dfrac{3}{2}x + 6$ , so $y = -\dfrac{3}{2}x + 4$ again. A line has many point-slope equations (one for each point) but only one slope-intercept equation. That makes slope-intercept form a good way to compare answers.

Point-slope form in context

Real data rarely hands you the starting value. More often you know two moments in time, and point-slope form turns them into a model.

Worked example: A burning candle

A candle burns at a steady rate. After $3$ hours it is $18$ cm tall; after $7$ hours it is $10$ cm tall. Write an equation for the height $h$ after $t$ hours, and find the candle's height before it was lit.

Slope. $m = \dfrac{10 - 18}{7 - 3} = \dfrac{-8}{4} = -2$ cm per hour.

Point-slope. Using $(3, 18)$ : $\;h - 18 = -2(t - 3)$ .

Simplify. $h - 18 = -2t + 6$ , so $h = -2t + 24$ .

Before it was lit, $t = 0$ , so $h = 24$ . The candle started at $24$ cm tall and loses $2$ cm each hour.

Choosing a form

you know	easiest start
slope and $y$ -intercept	$y = mx + b$
slope and any point	$y - y_1 = m(x - x_1)$
two points	find $m$ , then $y - y_1 = m(x - x_1)$

Whichever form you start with, you can always convert to slope-intercept form at the end.

Practice

Practice 1

Which is an equation of the line through $(2, 5)$ with slope $3$ ?

Practice 2

The line $y - 4 = -6(x + 1)$ passes through which point that you can read directly from the equation?

Enter a point like (2, -3)

Practice 3

What is the slope of the line $y + 3 = \dfrac{2}{5}(x - 8)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A line passes through $(-3, 1)$ with slope $2$ . Write its equation in slope-intercept form, $y = mx + b$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

Write the equation of the line through $(1, -4)$ and $(5, 8)$ in slope-intercept form.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 6

Write the equation of the line shown in slope-intercept form.

y = -2x + 1(-1, 3)(2, -3)Open in grapher →

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

Write the equation of the line through $(-6, 2)$ and $(3, -4)$ in slope-intercept form.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 8

A taxi's fare is a linear function of distance. A $4$ -mile ride costs $13.00 and a $10$ -mile ride costs $25.60. What does a $15$ -mile ride cost, in dollars?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.