Writing Equations of Lines – Summary Notes
In general … always use the equation y  mx  b .
Remember that m = slope
b = y-intercept (the point with coordinates (0, b ) )
and x and y represent any ( x , y ) point on the line.
Also remember that any time you are given two points you can find the slope using the
formula m 
y 2  y1
.
x2  x1
And, given an equation of a line, the slope is found by solving for y (because this will give
y  mx  b , and then you’ll see the slope as the coefficient of x).
The problem is complete when m and b have been calculated.
Problem A: Given the slope and the y-intercept
Step 1: Start with the general equation. Plug in all the given information.
Step 2: This problem is complete!
Example: Write the equation of the line in slope-intercept
1
form with slope =
and y-intercept of 6.
3
y  mx  b
y
1
x6
3
Problem B: Given the slope and any point on the line
Step 1: Start with the general equation. Plug in all the given information.
In this case it will be m , x , and y .
Step 2: Use your algebra skills to solve for b.
Step 3: Go back to the general equation and substitute in m and b.
Step 4: This problem is complete!
Example: Write the equation of the line in slope-intercept
3
form with slope =
and contains the point(-4, 5).
4
3
, x  4, y  5
4
y  mx  b
3
5  (4)  b
4
5  3  b
b8
3
y  x8
4
So, m 
Problem C: Given two points on the line
Step 1: Since you are given two points, you can use the formula to find slope.
Step 2: Now go to the general equation. Plug in the slope that you just
calculated (as m) and either one of the points as x and y . It doesn’t
matter which one you use – they will both yield the same answers.
Step 3: Use your algebra skills to solve for b.
Step 4: Go back to the general equation and substitute in m and b.
Step 5: This problem is complete!
Example: Write the equation of the line
in slope-intercept form that
contains the points (-4, 5) and (6, 10).
Find slope...
10  5
5 1
m


6  (4) 10 2
y  mx  b
I ' ll choose the point (6,10) as ( x, y )
1
(6)  b
2
10  3  b
10 
b7
y
1
x7
2
******* Facts about parallel and perpendicular lines. *******
 Parallel lines have equal slopes.
So, if a line has a slope of

7
7
, then every parallel line has a slope of
.
8
8
Perpendicular lines have slopes that are opposite reciprocals.
So, if a line has a slope of
7
8
, then every perpendicular line has a slope of .
7
8
Practice: Given the line 3x  2 y  8 , find the slope of (a) a parallel line, and
(b) a perpendicular line.
First: Solve for y to find the slope of
the given line.
3x  2 y  8
 3x
 3x
2 y   3x  8
2
2
3
y
x4
2
So, the slope of
the given line is
m
3
2
Second: a parallel line will have the
same slope; so, the slope of the
parallel line is m 
3
.
2
Third: a perpendicular line will have the
opposite reciprocal slope; so the
slope of the perpendicular line is m 
2
.
3
Problem D: Given a parallel line and a point on your new line
Step 1: Since you are given a parallel line, you know that your new line will
have the same slope.
Step 2: Solve for y to find the slope of the given line. (See above for help.)
Step 3: Now go to the general equation. Plug in the slope that you just
calculated (because parallel lines have the same slope) and the
coordinates of the point as x and y .
Step 4: Use your algebra skills to solve for b.
Step 5: Go back to the general equation and substitute in m and b.
Step 6: This problem is complete!
Example: Write the equation of the line
in slope-intercept form that is
parallel to the line 4 x  2 y  2
and contains the point (-3, 5).
Solve the given equation for y to find its slope;
4 x  2 y  2
 4x
 4x
 2 y  4 x  2
2
2
y  2x  1
So, m  2
Now use y  mx  b
5  2(3)  b
5  6  b
b  11
y  2 x  11
Problem E: Given a perpendicular line and a point on your new line
Step 1: Since you are given a perpendicular line, you know that your new line
will have the opposite reciprocal slope.
Step 2: Solve for y to find the slope of the given line. (See above for help.)
Step 3: Now go to the general equation. Plug in the opposite reciprocal of the
slope that you just calculated and the coordinates of the point as x and y .
Step 4: Use your algebra skills to solve for b.
Step 5: Go back to the general equation and substitute in m and b.
Step 6: This problem is complete!
See the work above for finding the slope
Example: Write the equation of the line
in slope-intercept form that is
perpendicular to the line
4 x  2 y  2 and contains the
point (-2, 5).
m  2, so new m 
Now use y  mx  b
5
1
(2)  b
2
5  1 b
b4
y
1
x4
2
1
2