# Writing Equations

### 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:

 y - k = m(x - h)

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)

Example 3: Write an equation of the line which is parallel to the line y = 3x + 2 and passes through (- 1, 2) .

m = 3 , h = - 1 , and k = 2 .
The equation of the line is y - 2 = 3(x + 1) .

Example 4: Write an equation of the line which is perpendicular to the line y - 8 = 2(x + 2) and passes through (7, 0) .

The slope is the opposite reciprocal of 2 : m = - . h = 7 and k = 0 .
The equation of the line is y - 0 = - (x - 7) , which is equivalent to y = - (x - 7) .

Example 5: Write an equation of the line with slope m = 4 that passes through the point (0, 3) .
m = 4 , h = 0 , and k = 3 .

The equation of the line is y - 3 = 4x. If we move -3 to the other side-- y = 4x + 3 --we get the equation in slope-intercept form.

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