## Point-Slope Form

Slope-intercept form is useful when we know the y- intercept of a line. However, we are not always given this information. When we know the slope and one point which is not the *y*-intercept, we can write the equation in point-slope form.

Equations in point-slope form look like this:

where

*m* is the slope of the line and

(*h*, *k*) is a point on the line (any point works).

To write an equation in point-slope form, given a graph of that equation, first determine the slope by picking two points. Then pick any point on the line and write it as an ordered pair (*h*, *k*). It does not matter which point you pick, as long as it is on the line--different points yield different constants, but the resulting equations will describe the same line.

Finally, write the equation, substituting numerical values in for *m*, *h*, and *k*. Check your equation by picking a point on the line--not the point you chose as (*h*, *k*)--and confirming that it satisfies the equation.

*Example 1*: Write an equation of the following line in point-slope form:

Graph of a Line

First, find the slope using the points

(- 2, 3) and

(3, - 1):

*m* = = = - .

Next, pick a point -- for example,

(- 2, 3). Using this point,

*h* = - 2 and

*k* = 3.

Therefore, the equation of this line is

*y* - 3 = - (*x* - (- 2)), which is equivalent to

*y* - 3 = - (*x* + 2).

Check using the point (3, -1):

-1 - 3 = - (3 + 2) ? Yes.

*Example 2*: Write an equation of the line which passes through (3, 4) and has slope *m* = 5.

*h* = 3 and *k* = 4. *y* - 4 = 5(*x* - 3)