Equations of a Line Review

You have learned about both the slope-intercept and the point-slope form of a line. You can use either of these forms to write the equation of any line, but each form gives you unique information about the line itself.

The boxes below review what you have learned about each of these forms of a line.

Slope-Intercept Form of a Line

The general equation is:

$ y = mx + b $.

In this equation:

$ y $ is the dependent variable
$ m $ is the line's slope
$ x $ is the independent variable
$ b $ is the $ y $-intercept

Point-Slope Form of a Line

The general equation is:

$ y - y_{1} = m(x - x_{1}) $.

In this equation:

$ y $ is the dependent variable
$ m $ is the line's slope
$ x $ is the independent variable
$ (x_{1},\ y_{1}) $ are the coordinates of an ordered pair that lies along the line

Recall that you can use algebraic techniques to rewrite any equation from point-slope form into slope-intercept form. For example:

Rewrite the equation $ y - 7 = \frac{5}{3}(x - 3) $ in slope-intercept form. Then, give the coordinates of the line's $ y $-intercept.

The steps for re-writing an equation in slope-intercept form and finding the location of the $ y $-intercept are shown in the table below. Click each step to see it applied to the example.

	This equation does not contain any double negatives.
	$ y - 7 = \frac{5}{3}\left( x - 3 \right) $ $ y - 7 = \frac{5}{3}x - 5 $
	$ y - 7 = \frac{5}{3}x - 5 $ $ y - 7 \ \require{color}\colorbox{yellow}{$ + \ 7 $} = \frac{5}{3}x - 5 \ \require{color}\colorbox{yellow}{$ + \ 7 $} $ $ y = \frac{5}{3}x + 2 $
	The value of $ b $ gives you the location of the $ y $-intercept. $ y = \frac{5}{3}x \ \require{color}\colorbox{yellow}{$ + \ 2 $} $ $ (0,\ 2) $

The value of an equation's $ y $-intercept tells you where the line crosses the $ y $-axis. The value of the $ x $-coordinate at the $ y $-intercept is always $ 0 $.

You have also learned how to solve for the location of a line's $ x $-intercept by substituting $ y = 0 $ into the equation and solving for $ x $. This will tell you where the line crosses the $ x $-axis. For example:

Where does the line $ y = \frac{5}{3}x + 2 $ cross the $ x $-axis?

How well do you remember how to work with the slope-intercept and the point-slope form of a line? Use the activity below to practice. Read each question and then select the best answer.

$ y = - \frac{3}{4}x + 5 $
$ y = - \frac{3}{4}x - 2 $
$ y = - \frac{3}{4}x - 10 $

Rewrite the equation by distributing on the right-hand side and then collecting the like terms. $ y - 2 = - \frac{3}{4}(x - 4) $ $ y - 2 = - \frac{3}{4}x + 3 $ $ y - 2 + 2 = - \frac{3}{4}x + 3 + 2 $ $ y= - \frac{3}{4}x + 5 $

Rewrite the equation by distributing on the right-hand side and then collecting the like terms.

$ \left( - \frac{3}{4},\ 5 \right) $
$ (0, - 5) $
$ (0,\ 5) $

You can rewrite this equation in slope-intercept form or recognize that the $ (x_{1},\ y_{1}) $ ordered pair given is the $ y $-intercept.

You can rewrite this equation in slope-intercept form or recognize that the $ (x_{1},\ y_{1}) $ ordered pair given is the $ y $-intercept. $ y - 5 = - \frac{3}{4}(x - 0) $ $ y - y_{1} = m(x - x_{1}) $ $ \left( x_{1},\ y_{1} \right) = \left( 0,5 \right) $

$ \left( 0, - \frac{3}{4} \right) $
$ \left( \frac{20}{3},\ 0 \right) $
$ \left( - \frac{20}{3}, - \frac{3}{4} \right) $

You can rewrite this equation in slope-intercept form, substitute $ y = 0 $ into the equation, and solve for $ x $. You can also recognize that the $ (x_{1},\ y_{1}) $ ordered pair given is the $ x $-intercept. $ y - 0 = - \frac{3}{4}(x - \frac{20}{3}) $ $ y - y_{1} = m(x - x_{1}) $ $ \left( x_{1},\ y_{1} \right) = \left( \frac{20}{3},\ 0 \right) $

$ y - 2 = - \frac{3}{4}(x - 2) $
$ y - 5 = - \frac{3}{4}(x - 6) $
$ y - \left( - 1 \right) = - \frac{3}{4}\left( x - 8 \right) $

You can rewrite each point-slope equation into slope-intercept form to see which one becomes $ y = - \frac{3}{4}x + 5 $. You could also use the given $ (x_{1},\ y_{1}) $ coordinates of each point-slope equation to see which one makes the slope-intercept equation true. $ y - ( - 1) = - \frac{3}{4}(x - 8) $ $ y + 1 = - \frac{3}{4}x + 6 $ $ y + 1 - 1 = - \frac{3}{4}x + 6 - 1 $ $ y= - \frac{3}{4}x + 5 $

Questions answered correctly:

Questions answered incorrectly:

	This equation does not contain any double negatives.
	\( y - 7 = \frac{5}{3}\left( x - 3 \right) \) \( y - 7 = \frac{5}{3}x - 5 \)
	\( y - 7 = \frac{5}{3}x - 5 \) \( y - 7 \ \require{color}\colorbox{yellow}{$ + \ 7 $} = \frac{5}{3}x - 5 \ \require{color}\colorbox{yellow}{$ + \ 7 $} \) \( y = \frac{5}{3}x + 2 \)
	The value of \( b \) gives you the location of the \( y \)-intercept. \( y = \frac{5}{3}x \ \require{color}\colorbox{yellow}{$ + \ 2 $} \) \( (0,\ 2) \)

Do you remember how to read and work with the different forms of a line's equation?

Let's Review

Slope-Intercept Form of a Line

Point-Slope Form of a Line

Show your skills!

Which expresses the equation \( y - 2 = - \frac{3}{4}(x - 4) \) in slope-intercept form?

What is the location of the \( y \)-intercept for the line \( y - 5 = - \frac{3}{4}(x - 0) \)?

Which ordered pair is the \( x \)-intercept of the equation \( y - 0 = - \frac{3}{4}\left( x - \frac{20}{3} \right) \)?

The equation \( y = - \frac{3}{4}x + 5 \) is the slope-intercept form of which point-slope equation?

Summary