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:
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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.
