Express your equation in slope-intercept form. The way you do this is by solving for y so that your equation looks like y = mx + b. This is “slope intercept form.” Let’s start with an example: 6x + 3y = 9. This equation is not in slope intercept form yet. To get there, we solve for y:
6x + 3y = 9 , we subtract 6x from both sides
3y = -6x + 9 , we need to solve for y so we divide both sides by 3
y = -2x + 3 , this is the equation in slope-intercept form!
Read off slope from the slope-intercept form equation. Honestly, this is really, really easy. Slope intercept form for a line is y = mx + b where m is the slope and b is the y-intercept. To find slope just read off what number is being multiplied by x and that is the slope. In our example, the slope is -2, or stated a different way, m = -2.
Bonus! You can also read off the y-intercept from the slope-intercept form equation. Remember that the y-intercept is just the ‘b’ in y = mx + b, so in our example b = 3. Just read off what constant is in the place of ‘b’ in your equations.
Know special cases for horizontal and vertical lines. For lines of the form x = constant there is no y-intercept and slope is undefined (aka infinity) because these lines are vertical lines.
For horizontal lines, their equations are of the form y = constant, so its y-intercept is equal to that constant and its slope is 0. (It’s like there is a zero multiplied by x, so there is no x term, there are zero x’s.)
Practice makes perfect. Here are some more examples to really hit this home
y = (1/2)x – 9 , slope = ½ , y-intercept = -9
y = -4 , slope = 0 , y-intercept = -4 (horizontal line)
x = -1 , slope is undefined, there is no y intercept (vertical line)
y = ax + np where a, n, and p are constants, slope = a, y-intercept = np