Write an equation for a parallel or perpendicular line

T.20 Write an equation for a parallel or perpendicular line by Delta publications

key notes:

Recap: Slope-Intercept Form

A line in slope-intercept form is:

y=mx+b

where:

  • m = slope
  • b = y-intercept

Parallel Lines

  • Rule: Parallel lines have the same slope.
    If line 1 has slope m, then the parallel line also has slope m.

Steps to write the equation:

  • Find the slope m from the given line.
  • Use the point you are given that lies on the new line.
  • Use point-slope form:

y−y1=m(x−x1)

Simplify to slope-intercept form y=mx+b

Example:
Write an equation of a line parallel to y=2x+5 passing through (3, -4).

  • Given slope m=2 (same as original line).
  • Use point (3, -4):

y−(−4)=2(x−3)

y+4=2x−6

y=2x−10

Perpendicular Lines

Rule: Perpendicular lines have slopes that are negative reciprocals:

m1×m2=−1

if m1​=a, then m2=−1/a

Steps to write the equation:

  1. Find the slope of the given line.
  2. Find the negative reciprocal of that slope for the new line.
  3. Use the given point and the new slope in point-slope form.
  4. Simplify to slope-intercept form.

Example:
Write an equation of a line perpendicular to y=1/2x+7 passing through (4, 1).

  • Given slope m1=1/2
  • Perpendicular slope m2=−2
  • Use point (4, 1):

y−1=−2(x−4)

y−1=−2x+8

y = -2x + 9

Special Cases

  • A horizontal line (m=0) is perpendicular to a vertical line (m=undefined).
  • A vertical line x=a parallel line will also be x=b (different constant).

Summary Table

Type of LineSlope RuleExample
ParallelSame slope my=3x+2, y=3x−5
Perpendicularm1×m2=−1m1=4,m2=-1/4

Practice Problems

  1. Write an equation of a line parallel to y=−3x+7 passing through (2, 5).
  2. Write an equation of a line perpendicular to y=5x−1 passing through (-4, 2).
  3. A line passes through (0, -3) and is parallel to y=2/5x+4. Write its equation.
  4. Find the equation of a line perpendicular to y=−3/4x+6 and passing through (1, 1).

Learn with an example

Line s has an equation of y = 3x + 4. Parallel to line s is line t, which passes
through the point (1, 6).

What is the equation of line t?

Step 1: Find the slope of line s.

First, find the slope m of line s. This is the only time you will use the equation of line s.

y = mx + b

y = –3x + 4

Line s has a slope m of 3.
Step 2: Find the slope of line t.
Line t is parallel to s, so its slope is the same: -3.
Step 3: Use the slope of line t and a point on line t to find its
y-intercept.
Plug the slope m = 3 and the point (1, 6) into the slope-intercept formula.
Then solve for the y-intercept b.
y = mx + b
6 = 3(1) + b Plugin y = 6, m = 3, and x = 1
6 = 3 + b Multiply
9 = b Add 3 to both sides
Line t has a y-intercept of 9.
Step 4: Use the slope of line t and the y-intercept of line t to find the
equation of the line.
Plug the slope m =- 3 and the y-intercept b = 9 into the slope-intercept
formula.
y = mx + b
y = 3x + 9 Plugin m = -3 and b = 9
The equation of line t in slope-intercept form is y = -3x + 9

Line k has an equation of y = 2x + 7. Line l, which is perpendicular to line k,
includes the point (2, 7).

What is the equation of line l?

Step 1: Find the slope of line k.
First, find the slope m of line k. This is the only time you will use the equation of line k.
y = mx + b
y = 2x + 7
Line k has a slope m of -2.

Step 2: Find the slope of line l.
Line l is perpendicular to k, so its slope is the opposite reciprocal: 1/2.
Step 3: Use the slope of line l and a point on line l to find its y-intercept.

Step 3: Use the slope of line l and a point on line l to find its y-intercept.

Plug the slope m=1/2 and the point(2, –7) into the slope-intercept formula. Then solve for they-intercept b.

y = mx + b

–7 =1/2 ( 2 )+b Plugin y= –7,m=1/2, and x= 2

–7 = 1 + b Multiply and simplify

–8 = b Subtract1 from both sides

Line l has ay-intercept of–8.

Step 4: Use the slope of line l and the y-intercept of line l to find the equation of the line.

Plug the slope m=1/2 and they-intercept b = –8 in to the slope-intercept formula.

y = mx + b

y =1/2 x+–8 Plugin m=1/2 and b= –8

y =1/2 x− 8
Rewrite as subtraction

The equation of line in slope-intercept form is y=1/2 x− 8

The equation of line g is y =1/7x − 8. Line h includes the point (9, 1) and is parallel to line g.

What is the equation of line h?

Step 1: Find the slope of line g.
First find the slope m of line g. This is the only time you will use the equation of
line g.
y = mx + b
y =1/7x − 8
Line g has a slope m of 1/7.

Step 2: Find the slope of line h.
Line h is parallel to g, so its slope is the same:1/7.

Step 3: Use the slope of line h and a point on line h to find its
y-intercept. Plug the slope m =1/7 and the point (9, 1) into the slope-intercept formula.
Then solve for the y-intercept b.
y = mx + b
1 =1/7(9) + b Plug in y = 1, m =1/7, and x = 9
1 =9/7+b Multiply = b Subtract 9/7 from both sides = b Rewrite with a common denominator
= b Subtract Line h has a y-intercept of -2/7.
Step 4: Use the slope of line h and the y-intercept of line h to find the
equation of the line.
Plug the slope m =1/7and the y-intercept b =-2/7 into the slope-intercept
formula.
y = mx + b
y =1/7x +-2/7 Plug in m =1/7 and b =-2/7
y =1/7x −2/7 Rewrite as subtraction


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