Using Point-Slope Form to Write the Equation of a Line

There's more than one way to write the equation of a straight line. You've probably worked with slope-intercept form ($y = mx + b$), which is great when you know the slope and the y-intercept. But what if you know the slope and some other point — not the y-intercept? That's exactly the situation point-slope form was made for.

The Formula

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

Here $(x_1, y_1)$ is any known point on the line, and $m$ is the slope. The subscript just labels your specific known coordinates — it doesn't mean multiply by 1.

Think of it this way: if you know where you're starting (a point) and which direction you're heading (the slope), you can draw exactly one line. Point-slope form captures that idea directly.

Writing an Equation from a Point and Slope

Example 1

Write the equation of the line through $(4, 3)$ with a slope of $2$.

Plug your values into the formula. Your point gives you $x_1 = 4$ and $y_1 = 3$, and the slope is $m = 2$:

$$y - 3 = 2(x - 4)$$

That's the answer in point-slope form. Depending on what the problem asks for, you might stop there — or you might convert it to slope-intercept form (more on that below).

Example 2

Write the equation of the line through $(-1, 5)$ with a slope of $\frac{1}{2}$.

Here $x_1 = -1$, $y_1 = 5$, and $m = \frac{1}{2}$:

$$y - 5 = \frac{1}{2}(x - (-1))$$

$$y - 5 = \frac{1}{2}(x + 1)$$

Watch the sign carefully. Subtracting a negative gives a positive — $x - (-1) = x + 1$. That's a spot where it's easy to slip up.

Reading an Equation from a Graph

Point-slope form works in reverse too. Given a graph, you can identify a point, calculate the slope, and write the equation.

Example 3

Find the equation of the line shown in this graph:

graph of a line passing through (-1, 0) with slope 2

You need two things: a point and the slope.

Finding a point: Choose any location where the line lands cleanly on a grid intersection. The point $(-1, 0)$ is an easy read — it sits right on the x-axis.

Finding the slope: Pick two clear grid points and count. Using the triangle visible on the graph, the line rises 2 units for every 1 it runs to the right. So $m = 2$.

Plug in with $x_1 = -1$, $y_1 = 0$, $m = 2$:

$$y - 0 = 2(x - (-1))$$

$$y = 2(x + 1)$$

Any point on the line would have given the same result — the choice of $(-1, 0)$ just made the arithmetic clean.

Converting to Slope-Intercept Form

Point-slope form and slope-intercept form describe the same line — they're just arranged differently. To convert, distribute and solve for $y$.

Starting from Example 1:

$$y - 3 = 2(x - 4)$$ $$y - 3 = 2x - 8$$ $$y = 2x - 5$$

Now you can read the y-intercept directly: the line crosses the y-axis at $-5$.

From Example 2:

$$y - 5 = \frac{1}{2}(x + 1)$$ $$y - 5 = \frac{1}{2}x + \frac{1}{2}$$ $$y = \frac{1}{2}x + 5\frac{1}{2}$$

Practice Problems

Write the equation (in point-slope form) of the line through $(2, -3)$ with slope $4$.

Show answer$y - (-3) = 4(x - 2)$, which simplifies to $y + 3 = 4(x - 2)$

Write the equation of the line through $(0, 6)$ with slope $-\frac{2}{3}$.

Show answer$y - 6 = -\frac{2}{3}(x - 0)$, which simplifies to $y - 6 = -\frac{2}{3}x$. Since the known point is the y-intercept, this is also $y = -\frac{2}{3}x + 6$.

Convert $y + 4 = 3(x - 2)$ to slope-intercept form.

Show answerDistribute: $y + 4 = 3x - 6$, then subtract 4: $y = 3x - 10$

A line passes through $(1, 3)$ and $(4, 9)$. Write its equation in point-slope form.

Show answerFirst find the slope: $m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$. Using $(1, 3)$: $y - 3 = 2(x - 1)$. Using $(4, 9)$ gives $y - 9 = 2(x - 4)$ — both are correct forms of the same line.

A line has the equation $y - 1 = -\frac{1}{2}(x + 6)$. What is the slope, and what point does the equation tell you is on the line?

Show answerThe slope is $m = -\frac{1}{2}$. Matching the form $y - y_1 = m(x - x_1)$ gives $x_1 = -6$ and $y_1 = 1$, so the point is $(-6, 1)$.