The Slope-Intercept Form
The slope-intercept form of the equation of a line is probably the most all-around useful version. It lets you pick out the slope and the y-intercept just by looking at it.
Here's what this form looks like. If m is the line's slope and b is its y-intercept then the line's equation is:
y = mx + b
Here's what makes this form so magical. If I tell you give you the equation of a line, all you have to do is solve it for y and you've got all the information you need to graph it at your fingertips. For example, if I gave you the equation
y = -2x + 3
Then you know that the line's slope is -2 and its y-intercept is 3.
Going in the other direction, if I tell you that the slope of a line is -2 and its y-intercept is -4 then you can get the equation just by substituting -2 for m and -4 for b in the slope-intercept form. That would give you this as the line's equation:
y = -2x - 4
A slightly more complicated example, that we'll look at in detail in the examples below, involves taking two points and using this form to find the line's equation. Here are the steps you'll need to follow:
- Calculate the slope of the line. (See the "Properties" section of the classroom if you need to know the details.)
- Substitute the number you calculated in step (1) for the m in the slope-intercept form.
- If the x-coordinate of one of the points that you were given is 0 then the y value of that point is the y-intercept. Substitute that number for the b in the slope-intercept form and you're done. Otherwise go to step four.
- Pick one of your points, it doesn't matter which, and substitute it into the equation you made in step (2). Solve that equation for b.
- Substitute the value you found in step (4) into the equation you made in step (2) and you're done.
Example 1
Find the equation of the line through the points (2, 7) and (0, 3).
The first thing we need to do is find the line's slope.
$$m=\frac{7-3}{2-0}=\frac{4}{2}=2$$Now we can substitute that into the slope-intercept form of the equation and we're half way there.
y = 2x + b
Now, if you look closely at the second point, you'll see that it's on the y-axis. That means that 3 must be our line's y-intercept. If we substitute that for b in our "half-way" equation and we're done.
y = 2x + 3
Example 3
Find the equation of the line through the points (5, 3) and (7, 3).
Just like in the previous examples, the first thing we need to do is find the line's slope.
$$m=\frac{3-3}{5-7}=\frac{0}{2}=0$$Don't let the zero throw you. Our method is still going to work. If we substitute 0 into the slope-intercept form of the equation, we'll get:
y = 0 · x + b = 0 + b = b
Now, I'll take the second point (again it doesn't matter which you chose) and substitute it's y value into our partial equation.
b = 3
Putting that back into our "half-way" equation gives us our final answer.
y = 3
Example 2
Find the equation of the line through the points (3, 5) and (-1, 7).
The first thing we need to do is find the line's slope.
$$m=\frac{5-7}{3-(-1)}=\frac{-2}{4}=-\frac{1}{2}$$Now we can substitute that into the slope-intercept form of the equation and we're half way there.
$$y = -\frac{1}{2}x + b$$Now all we have to do is figure out what b equals. Unfortunately, neither of our points are where the line crosses the y-axis so we can't get the y-intercept from them. That means we'll have to use our equation to solve for it. I'm going to use the first point (3, 5) to do this although they both work just as well. I'll start by substituting 3 for x and 5 for y in our "half-way" equation.
$$5=-\frac{1}{2} \cdot 3 + b = -\frac{3}{2}+ b$$ $$b = 5 + \frac{3}{2} = \frac{10}{2} + \frac{3}{2} = \frac{13}{2}$$If we substitute 13/2 into our "half-way" equation for b, we'll have our final answer.
$$y=-\frac{1}{2}x + \frac{13}{2}$$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
Individual Examples
Dyanmic Tutorial - Using the Slope-Intercept 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.
