chapter 3 introduction to graphing and equations of lines section 6 parallel and perpendicular lines
TRANSCRIPT
Chapter 3
Introduction to Graphing and Equations of
Lines
Section 6
Parallel and Perpendicular Lines
3.6 - 2Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Section 3.6 Objectives
1 Determine Whether Two Lines Are Parallel
2 Find the Equation of a Line Parallel to a Given
Line
3 Determine Whether Two Lines Are
Perpendicular
4 Find the Equation of a Line Perpendicular to a
Given Line
3.6 - 3Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Parallel Lines
Two nonvertical lines are parallel if and only if their slopes are equal and they have different y-intercepts. Vertical lines are parallel if they have different x-intercepts.
m1 = m2
x
y
m1 = m2
x
y
3.6 - 4Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Parallel Lines
Example:Determine whether the line 6x + 2y = 9 is parallel to – 3x – y = 3.
Find the slope of each line.
6x + 2y = 9
2 6 9y x=− +
293y x=− +
1 3m =−
– 3x – y = 3
3 3y x− = +
3 3y x=− −
2 3m =−
The slopes are the same so the lines are parallel.
3.6 - 5Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Parallel Lines
Example:Find the equation of the line that is parallel to 4x + y = – 8 and contains the point (2, – 3). Write the equation in slope-intercept form.
Find the slope of the line.4x + y = – 8
4m =−y = – 4x – 8
Use the point-slope form to find the equation.
1 1( )y y m x x− = −( ) ( )3 4 2y x− −− = −
3 4 8y x+ =− +
4 5y x=− +
Simplify.
Subtract 3 from both sides.
Substitute in the values.
3.6 - 6Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Perpendicular Lines
Two nonvertical lines are perpendicular if and only if the product of their slopes is – 1. Any vertical line is perpendicular to any horizontal line.
m1m2 = 1
x
y
or
m1 = 2
1m
−The slopes are negative reciprocals of each other.
3.6 - 7Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Perpendicular Lines
Example:Determine whether the line x + 3y = – 15 is perpendicular to– 3x + y = – 1 .
Find the slope of each line.
x + 3y = – 15
3 15y x=− −13
5y x=− −
113
m =−
– 3x + y = – 1
3 1y x= −2 3m =
The slopes are negative reciprocals so the lines are perpendicular.
3.6 - 8Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Perpendicular LinesExample:Find the equation of the line that is perpendicular to the line – 2x + 5y = 3 and contains the point (2, – 3). Write the equation in slope-intercept form.
Find the slope of the line.
– 2x + 5y = 3
25
m =
5y = 2x + 3
Continued.
2 35 5
y x= +
3.6 - 9Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra
Perpendicular Lines
Example continued:
1 1( )y y m x x− = −
( ) 53 22
( )y x− =− −−
53 52
y x+ =− +
5 22
y x=− +
Simplify.
Subtract 3 from both sides.
Use the negative reciprocal slope.
Use the point-slope form to find the equation.