Solved Examples and Worksheet for Writing Linear Equations of Parallel and Perpendicular Lines

Q1Which equation of the line is parallel to y = - 2x + 4 and passes through the point (2, 4)?

A. y = - 12x - 8
B. y = - 2(x - 8)
C. y = - 2x + 8
D. y = 12x + 8

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = - 2x + 4 is - 2.
  [Compare with y = mx + b.]
Step: 3
The slope of the line parallel to the original line is - 2.
  [Parallel lines have same slope.]
Step: 4
The equation of the line with slope - 2 in slope-intercept form is y = - 2x + b.
Step: 5
Substitute the point (2, 4) in the above equation.
Step: 6
4 = - 2(2) + b
  [Replace x with 2 and y with 4.]
Step: 7
b = 8
  [Solve for b.]
Step: 8
The y-intercept is b = 8.
Step: 9
y = - 2x + 8
  [Replace m with - 2 and b with 8 in y = mx + b.]
Step: 10
The equation of the line parallel to y = - 2x + 4 and passing through the point (2, 4) is y = - 2x + 8.
Correct Answer is :   y = - 2x + 8
Q2Which equation of the line is parallel to y = - 35x + 5 and passes through the point (- 2, 6)?
A. y = - 35x + 245
B. y = 35x - 245
C. y = - 35x + 365
D. y = 35x + 245

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
The slope of the line y = - 35x + 5 is - 35
  [Compare with the slope intercept form equation.]
Step: 3
The slope of line parallel to y = - 35x + 5 is - 35.
  [Parallel lines have equal slopes.]
Step: 4
The equation of the line with slope - 35 in slope-intercept form is y = - 35x + b
Step: 5
Substitute the point (- 2, 6) in the above equation.
Step: 6
6 = - 35(- 2) + b
  [Replace x with - 2 and y with 6]
Step: 7
b = 245
  [Solve for b.]
Step: 8
y = - 35x + 245
  [Replace m with - 35 and b with 245.]
Step: 9
The equation of the line parallel to y = - 35x + 5 is y = - 35x + 245which is passing through the point (- 2, 6).
Correct Answer is :   y = - 35x + 245
Q3Which equation of the line is parallel to y = 4x + 7 and passes through the point (3, 22)?
A. y = - 4x + 10
B. y = 4x + 10
C. y = - 4x - 10
D. y = 4x - 10

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 4x + 7 is 4.
  [Compare with y = mx + b.]
Step: 3
The slope of the line parallel to y = 4x + 7 is 4.
  [Parallel lines have same slope.]
Step: 4
The equation of the line with slope 4 in slope-intercept form is y = 4x + b.
Step: 5
Substitute the point (3, 22) in the above equation.
Step: 6
22 = 4(3) + b
  [Replace x with 3 and y with 22.]
Step: 7
b = 10
  [Solve for b.]
Step: 8
The y-intercept is b = 10.
Step: 9
y = 4x + 10
  [Replace m with 4 and b with 10 in y = mx + b.]
Step: 10
The equation of the line parallel to y = 4x + 7 and passing through the point (3, 22) is y = 4x + 10.
Correct Answer is :   y = 4x + 10
Q4Choose an equation of the line passing through the point (- 2, - 4) and perpendicular to the line y = 4x + 5.
A. y = 14(x - 2)
B. y + 4 = - 14(x + 2)
C. y + 4 = 14(x + 2)
D. y = - 14(x + 2)

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 4x + 5 is 4.
  [Compare with the equation in step 1.]
Step: 3
Slope of the line perpendicular to y = 4x + 5 is - 14.
  [Product of slopes of perpendicular lines is - 1.]
Step: 4
The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).
Step: 5
y - (- 4) = - 14[x - (- 2)]
  [Substitute (x1, y1) = (- 2, - 4) and m = - 14 in the equation in step 4.]
Step: 6
y + 4 = - 14(x + 2)
  [Simplify the equation.]
Step: 7
The equation of the line passing through the point (- 2, - 4) is y + 4 = - 14(x + 2).
Correct Answer is :    y + 4 = - 14(x + 2)
Q5Choose the equation of the line passing through the point (2, 3) and perpendicular to the line y = 3x + 4.

A. y - 3 = - 13(x - 2)
B. y - 3 = - 13(x + 2)
C. y + 3 = - 13(x - 2)
D. y - 3 = 13(x - 2)

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 3x + 4 is 3.
  [Compare with the equation in step 1.]
Step: 3
Slope of the line perpendicular to y = 3x + 4 is - 13 .
  [Product of slopes of perpendicular lines is - 1.]
Step: 4
The equation of the line passing through the point (x1, y1) with slope 'm' in point-slope form is y - y1 = m(x - x1).
Step: 5
Point (x1, y1) = (2, 3) and slope m of the perpendicular line = - 13.
Step: 6
y - 3 = - 13(x - 2)
  [Substitute x1 = 2, y1 = 3 and m = - 13 in the equation in step 4.]
Step: 7
The equation of the line passing through the point (2, 3) is y - 3 = - 13(x - 2).
Correct Answer is :   y - 3 = - 13(x - 2)
Q6Select the equation of the line passing through the point (- 3, 5) and perpendicular to the line y = 3x + 8.

A. y + 5 = 13(x + 3)
B. y - 5 = 13(x + 3)
C. y + 5 = - 13(x + 3)
D. y - 5 = - 13(x + 3)

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 3x + 8 is 3.
  [Compare with the equation in step 1.]
Step: 3
Slope of the line perpendicular to y = 3x + 8 is - 13 .
  [Product of slopes of perpendicular lines is - 1.]
Step: 4
The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).
Step: 5
Point (x1, y1) = (- 3, 5) and slope m of the perpendicular line = - 13
Step: 6
y - 5 = - 13(x - (- 3))
  [Substitute x1 = - 3, y1 = 5 and m = - 13 in the equation in step 4.]
Step: 7
y - 5 = - 13(x + 3)
  [Simplify the equation.]
Step: 8
So, the equation of the line passing through the point (- 3, 5) is y - 5 = - 13(x + 3).
Correct Answer is :   y - 5 = - 13(x + 3)
Q7Find the equation of line passing through (5, 6) and parallel to line x + 4y - 9 = 0.

A. x + 4y - 29 = 0
B. y + 4x - 9 = 0
C. x + y - 9 = 0
D. y + 4x - 29 = 0

Solutions
Share 
Step: 1
The slope of the line represented by x + 4y - 9 = 0 is - 14
  [Convert the equation to slope - intercept form and find the slope.]
Step: 2
Let L be the required line parallel to the above line.
Step: 3
Slope of the line L = m = slope of the given line = - 14
  [The slopes of parallel lines are equal.]
Step: 4
Since line L is passing through A(5, 6), the equation of line L is y - y1 = m(x - x1)
  [Use slope - point form of a line.]
Step: 5
y - 6 = - 14 (x - 5)
Step: 6
4y - 24 = - x + 5
Step: 7
x + 4y - 29 = 0
Correct Answer is :   x + 4y - 29 = 0
Q8Find the equation of line passing through (2, 5) and parallel to line x + 4y - 24 = 0.
A. x + 4y - 22 = 0
B. x + 4y - 18 = 0
C. x + 4y + 22 = 0
D. x + 4y + 18 = 0

Solutions
Share 
Step: 1
The slope of the line represented by x + 4y - 24 = 0 is - 14.
  [Convert the equation to slope - intercept form and find the slope.]
Step: 2
Let L be the required line parallel to the above line.
Step: 3
Slope of the line L = m = slope of the given line = - 14
  [The slopes of parallel lines are equal.]
Step: 4
Since line L is passing through A(2, 5), the equation of the line L is y - y1 = m(x - x1)
  [Use the slope - point form of line.]
Step: 5
y - 5 = - 14 (x - 2)
Step: 6
4y - 20 = - x + 2
Step: 7
x + 4y - 22 = 0
Correct Answer is :   x + 4y - 22 = 0
Q9Which equation of the line is parallel to y = - 47x + 7 and passes through the point (- 3, 6)?
A. y = - 47x + 547
B. y = 47x - 307
C. y = 47x + 307
D. y = - 47x + 307

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
The slope of the line y = - 47x + 7 is - 47
  [Compare with the slope intercept form equation.]
Step: 3
The slope of line parallel to y = - 47x + 7 is - 47.
  [Parallel lines have equal slopes.]
Step: 4
The equation of the line with slope - 47 in slope-intercept form is y = - 47x + b
Step: 5
Substitute the point (- 3, 6) in the above equation.
Step: 6
6 = - 47(- 3) + b
  [Replace x with - 3 and y with 6]
Step: 7
b = 307
  [Solve for b.]
Step: 8
y = - 47x + 307
  [Replace m with - 47 and b with 307.]
Step: 9
The equation of the line parallel to y = - 47x + 7 is y = - 47x + 307 which is passing through the point (- 3, 6).
Correct Answer is :   y = - 47x + 307
Q10Which equation of the line is parallel to y = 2x + 3 and passes through the point (1, 11)?
A. y = - 2x - 9
B. y = 2x + 9
C. y = 2x - 9
D. y = - 2x + 9

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 2x + 3 is 2.
  [Compare with y = mx + b.]
Step: 3
The slope of the line parallel to y = 2x + 3 is 2.
  [Parallel lines have same slope.]
Step: 4
The equation of the line with slope 2 in slope-intercept form is y = 2x + b.
Step: 5
Substitute the point (1, 11) in the above equation.
Step: 6
11 = 2(1) + b
  [Replace x with 1 and y with 11.]
Step: 7
b = 9
  [Solve for b.]
Step: 8
The y-intercept is b = 9
Step: 9
y = 2x + 9
  [Replace m with 2 and b with 9 in y = mx + b.]
Step: 10
The equation of the line parallel to y = 2x + 3 and passing through the point (1, 11) is
y = 2x + 9.
Correct Answer is :   y = 2x + 9
Q11Choose an equation of the line passing through the point (1, 5) and perpendicular to the line 2y = - 3x + 6.

A. y - 5 = 23 (x - 1)
B. y + 5 = 33 (x - 1)
C. y - 5 = 23 (x + 1)
D. y - 5 = 32 (x - 1)

Solutions
Share 
Step: 1
The given equation is 2y = - 3x + 6.
Step: 2
⇒ y = - 32x + 3
  [Writing the given equation in slope-intercept form y = mx + b.]
Step: 3
Slope of the line y = - 32x + 3 is - 32.
  [Where m is the slope.]
Step: 4
So, slope of the line which is perpendicular to the line 2y = - 3x + 6 is 23.
  [Product of the slopes = - 1.]
Step: 5
Now write the line which is perpendicular to the given line by using slope and the given point.
Step: 6
The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).
Step: 7
⇒ y - 5 = 23(x - 1)
  [Substitute x1 = 1 and y1 = 5 and m = 23 because the line should pass through the point (1, 5).]
Step: 8
∴ The equation of the line passing through the point (1, 5) and perpendicular to the line 2y = - 3x + 6 is y - 5 = 23(x - 1).
Correct Answer is :    y - 5 = 23 (x - 1)
Q12Choose the equation of the line passing through the point (- 4, - 4) and perpendicular to the line 3y = x - 3.
A. y + 4 = - 3 (x - 4)
B. y + 4 = - 3 (x + 4)
C. y + 4 = 3 (x + 4)
D. y - 4 = - 3 (x + 4)

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope, m and y-intercept, b is y = mx + b.
Step: 2
Slope of the line 3y = x - 3 is y = 13.
  [Compare with the equation in step 1.]
Step: 3
Slope of the line perpendicular to 3y = x - 3 is - 3.
  [Product of slopes of perpendicular lines is - 1.]
Step: 4
The equation of the line passing through the point (x1, y1) with slope ′m′ in point-slope form is y - y1 = m(x - x1).
Step: 5
Point (x1, y1) = (- 4, - 4) and slope m of the perpendicular line = - 3.
Step: 6
y + 4 = - 3(x + 4)
  [Substitute x1 = - 4, y1 = - 4, and m = - 3 in the equation of step 4.]
Step: 7
Therefore, the equation of the line passing through the point (- 4, - 4) is y + 4 = - 3(x + 4).
Correct Answer is :   y + 4 = - 3 (x + 4)
Q13Choose the equation of the line passing through the point (4, 6) and perpendicular to the line y = 8x + 14.

A. y = - 18(x) - 112
B. y = - 18(x) - 132
C. y = - 18(x) + 132
D. y = - 18(x) + 112

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
Slope of the line y = 8x + 14 is 8.
  [Compare with the equation in step 1.]
Step: 3
Slope of the line perpendicular to y = 8x + 14 is - 18.
  [Product of slopes of perpendicular lines is - 1.]
Step: 4
The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).
Step: 5
y - (6) = - 18[x - (4)]
  [Substitute (x1, y1) = (4, 6) and m = - 18 in the equation in step 4.]
Step: 6
y - 6 = - 18(x - 4)
  [Simplify the equation.]
Step: 7
y - 6 = - 18(x) + 12
Step: 8
y = - 18(x) + 132
Step: 9
The equation of the line passing through the point (4, 6) is y = - 18(x) + 132
Correct Answer is :   y = - 18(x) + 132
Q14What is the equation of the line in point-slope form that is parallel to y = - 4x + 7 and passes through the point (4, 6)?
A. y = - 4x - 22
B. y = - 4x + 22
C. y = - 4x + 10
D. y = - 4x - 10

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
y = - 4x + 7
  [Equation of the line given.]
Step: 3
Slope of the line y = - 4x + 7 is - 4.
  [Compare with the equation in step 1.]
Step: 4
The slope of the line parallel to the line y = - 4x + 7 is - 4.
  [Parallel lines have same slope.]
Step: 5
The required line passes through the point (4, 6) and has a slope, m = - 4.
Step: 6
The equation of the line passing through the point (x1, y1) with slope m is y - y1 = m(x - x1).
  [Point-slope form.]
Step: 7
y - 6 = - 4(x - 4)
  [Substitute x1 = 4, y1 = 6 and m = - 4 in the equation in step 6.]
Step: 8
y - 6 = - 4x + 16
Step: 9
y = - 4x + 22
Step: 10
The equation of the line in point-slope form is y = - 4x + 22
Correct Answer is :   y = - 4x + 22
Q15What is the equation of the line in point-slope form that is parallel to y = 45x + 6 and passes through the point (3, 5)?
A. y = 45(x) - 135
B. y = 45(x) - 375
C. y = 45(x) + 375
D. y = 45(x) + 135

Solutions
Share 
Step: 1
The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.
Step: 2
The equation is y = 45 x + 6.
Step: 3
Comparing the equation with equation in step 1, the slope of the line is 45.
Step: 4
So, the slope of line parallel to the line is 45.
  [Slope of parallel lines are same.]
Step: 5
The parallel line passes through the point (x1, y1) = (3, 5) and has a slope m = 45.
Step: 6
The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).
Step: 7
y - 5 = 45(x - 3)
  [Substitute x1 = 3, y1 = 5 and m = 45 in the equation in step 6.]
Step: 8
y - 5 = 45(x) - 125
Step: 9
y = 45(x) + 135
Step: 10
The equation of the parallel line in point-slope form is y = 45(x) + 135
Correct Answer is :   y = 45(x) + 135

HighSchool Math