The Point-Slope Form

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A Short Explanation

The point-slope form is similar to the slope-intercept form. What makes it special is that it works for any point on the line where, to use the slope-intercept form, the point has to be the y-intercept.

Here's what this form looks like. If m is the line's slope and (x₁, y₁) is a point on the line then the line's equation is:

y - y₁ = m(x - x₁)

Don't be confused by the subscripts. They're just there to distinguish the x and y values of the points that you know from the x and y variables in the equation. If were to write it out using words the equation would look like:

y - (the y coordinate of your point) = m(x - (the x coordinate of your point))

Using the point-slope equation is almost always just substitution and then some simplification.

Examples

Example 1

Find the equation of the line through the point (3, 6) whose slope is -2.

We're already given the slope and the point so all we need to do is substitute the values into the point-slope equation.

$$y - y_1 = m(x - x_1$$ $$y - 6 = -2(x - 3)$$ $$y - 6 = -2x + 6$$ $$y = -2x + 12$$

Example 3

Find the equation of the line through the points (5, 3) and (7, 3).

Since we aren't given the slope, the first thing we'll need to do is calculate it.

$$m = \frac{3 - 3}{5 - 7} = \frac{0}{-2} = 0$$

If we substitute that value and one of our points (it doesn't matter which) into the point-slope form of the equation we'll get our answer.

y - 3 = 0 · (x - 5)
y - 3 = 0
y = 3

Example 2

Find the equation of the line through the points (2, 6) and (-1, 7).

The first thing we need to do is find the line's slope.

$$m = \frac{6 - 7}{2 - (-1)} = -\frac{1}{3}$$

Now we can substitute the slope and the coordinates of one of our points (it doesn't matter which) into the point-slope equation and simplify it.

$$y - 6 = -\frac{1}{3}(x - 2)$$ $$y - 6 = -\frac{1}{3} x + \frac{2}{3}$$ $$y = -\frac{1}{3} x + \frac{2}{3} + 6$$ $$y = -\frac{1}{3} x + \frac{2}{3} + \frac{18}{3}$$ $$y = -\frac{1}{3} x + \frac{20}{3}$$

Quick Tip - Checking Your Answer

Everyone likes to have a way to check their answers. With the equation of a line, you can do this by taking both of your points and substituting them into the equation. If you get a true statement both times, e.g. 3 = 3 then you know you found the right equation. On the other hand, if you get a false statement like 3 = 2 then you'll know you made a mistake somewhere.

Video Lectures

Lectures

The Slope-Intercept Equation

In all of our previous discussions, we've started with the equation of a line and answered questions about it. In this lecture, we'll start by seeing how to find the slope and intercept given an equation that we'll see how we can use the slope-intercept to find the equation of a line given information about it. (lecture slides)

Individual Examples

Finding the Equation of a Line

In this example, we'll look at how to find the equation of a line using the slope-intercept equation when we don't know the line's y-intercept.

Dyanmic Tutorial - Using the Point-Slope Form

Directions: This solution has 4 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.