11.
Write the equation of a line perpendicular to the line $y$ = 3$x$ - 8 and having $y$ - intercept 2.
Solution:
Two lines are perpendicular if and only if the product of their slopes is - 1.
y = 3x - 8
[Equation of the line is in the form of y = mx + b, where m is slope and b is the y - intercept.]
Slope of the given line = 3
Then the slope of the line which is perpendicular to the given line = - 1 / 3
[3 × - 1 / 3 = - 1.]
y - intercept of the line which is perpendicular to the given line =2.
The equation of the line which is perpendicular to given line is y = - 1 / 3 x + 2
3y = - x + 6
[Multiply the equation by 3.]
So, the equation of the line which is perpendicular to the given line is x + 3y - 6 = 0.
Correct answer : (4)
12.
Find the equation of a line parallel to the line 5$y$ - 4$x$ = 7 and having $y$ - intercept - 3.
Solution:
Two lines are parallel if and only if their slopes are equal.
5y - 4x = 7
[Given equation of the line.]
y = 4 / 5x + 7 / 5
[Write the equation in slope - intercept form.]
Slope of the line = 4 / 5
Slope of the line which is parallel to given line = 4 / 5
y - intercept of the line which is parallel to given line = - 3.
The equation of the line which is parallel to given line is y = 4 / 5 x - 3
5y - 4x = - 15
[Multiply both sides with 5.]
So,the equation of the line which is parallel to given line is 5y - 4x = - 15.
Correct answer : (5)
13.
Find the slope of the lines that are ($a$) parallel and ($b$) perpendicular to the line $y$ = - 5$x$ + 9.
Solution:
The slope - intercept form of a line is y = mx + b, where m is slope and b is y - intercept.
y = - 5x + 9 is in the form of y = mx + b.
Slope, m = - 5.
Two lines are parallel if and only if their slopes are equal.
Slope of the parallel line is - 5.
Two lines are perpendicular if the product of their slopes is - 1.
Slope of the perpendicular line is 1 / 5.
The slope of any line parallel to the line y = - 5x + 9 is - 5 and the slope of any line perpendicular to the line y = - 5x + 9 is 1 / 5.
Correct answer : (1)
14.
Find the slope of the lines that are ($a$) parallel and ($b$) perpendicular to the line - $\frac{5}{4}$$x$ + $y$ = 7.
Solution:
The slope - intercept form of a line is y = mx + b, where m is slope and b is y - intercept.
- 5 / 4x + y = 7
[Given equation of the line.]
y = 7 + 5 / 4x
[Add 5 / 4x on both sides.]
y = 5 / 4x + 7
[Rewrite the equation in the form of y = mx + b.]
The slope of the equation is 5 / 4.
Two lines are parallel if and only if their slopes are equal.
So, the slope of the parallel line is 5 / 4.
The two lines are perpendicular if and only if the product of their slopes is - 1.
So, the slope of the perpendicular line is - 4 / 5.
The slope of the parallel line is 5 / 4 and the slope of the perpendicular line is - 4 / 5.
Correct answer : (5)
15.
Find the slopes of the lines that are ($a$) parallel and ($b$) perpendicular to the line $y$ = - 7.
Solution:
The slope - intercept form of the equation is y = mx + b, where m is slope and b is y - intercept.
y = - 7
[Given equation of the line.]
Slope of the given line is 0
The slope of the parallel line is same as the slope of the equation of any line.
So, the slope of the parallel line is 0.
The slope of the perpendicular line is negative reciprocals of the slope of any line.
So, the slope of the perpendicular line is undefined.
So, the slope of the parallel line is 0 and slope of the perpendicular line is undefined.
Correct answer : (2)
16.
Write an equation for the line in slope - intercept form that passes through (- 5, 3) and is parallel to 8$x$ + 3$y$ = 4.
Solution:
8x + 3y = 4
[Equation of the line.]
y = - 8 / 3 x + 4 / 3
[Rearrange in the form of y = mx + b.]
Slope of the line, m = - 8 / 3
Slope of the parallel line, m_{1} = - 8 / 3
[Slopes of parallel lines are equal.]
Equation of the line with slope m_{1} and y - intercept b is y = m_{1}x + b = - 8 / 3 x + b
If this line passes through the point (- 5, 3) then, 3 = - 8 / 3 (- 5) + b
b = - 31 / 3
[Solve for b.]
y = - 8 / 3 x - 31 / 3
[Substitute b in step 5.]
So, the equation of the line passes through (- 5, 3) and parallel to 8x + 3y = 4 is y = - 8 / 3x - 31 / 3.
Correct answer : (1)
17.
Write the equation of a line in slope - intercept form that passes through (- 2, 4) and perpendicular to - 3$x$ + $y$ = 5.
Solution:
- 3x + y = 5
[Original equation.]
y = 5 + 3x
[Add 3x on both sides.]
y = 3x + 5
[y = mx + b.]
The equation y = 3x + 5 is in the form of y = mx + b, where slope m = 3 and y - intercept b = 5.
The slope of the perpendicular line is negative reciprocal of the slope of the line.
So, the slope of perpendicular line is - 1 / 3.
Equation of the line that passes through (- 2, 4) with slope m = - 1 / 3 is
y - 4 = - 1 / 3(x + 2)
[Use y - y_{1} = m(x - x_{1}).]
y = - x3 - 2 / 3 + 4
[Add 4 on both sides.]
y = - x3 + 10 / 3
[Simplify.]
The slope - intercept form of the line that passes through (- 2, 4) and perpendicular to - 3x + y = 5 is y = - x3 + 10 / 3.
Correct answer : (4)
18.
Write an equation for the line in point - slope form that passes through (- 3, 2) and is parallel to - 2$x$ + 5$y$ = 0.
Solution:
- 2x + 5y = 0
[Given equation of the line.]
5y = 2x
[Add 2x on both sides.]
y = 2 / 5x
[Divide by 5 on both sides.]
The equation y = 2 / 5x is in the form of y = mx + b, where slope m = 2 / 5 and y - intercept b = 0.
Slope of the line parallel to - 2x + 5y = 0 is 2 / 5
[Parallel lines slopes are equal.]
The point - slope form of an equation that passes through (- 3, 2) with slope m = 2 / 5is
y - 2 = 2 / 5(x + 3)
[Use y - y_{1} = m(x - x_{1}).]
So, the equation of the line passing through the point (- 3, 2) and parallel to - 2x + 5y = 0 is y - 2 = 2 / 5(x + 3).
Correct answer : (2)
19.
Find the slopes of the lines that are ($a$) parallel and ($b$) perpendicular to the line $x$ = - 7.
Solution:
x = - 7
[Original equation.]
This is a vertical line equation.
The slope for vertical line is undefined.
The slope for parallel line is same as the slope of given line.
So, the slope of the line parallel to x = - 7 is undefined.
The slope of the perpendicular line is negative reciprocal of the slope of given line.
So, the slope of the line perpendicular to x = - 7 is 0.
Correct answer : (4)
20.
Write an equation for the line in slope - point form that passes through (6, - 5) and is perpendicular to 5$x$ - 2$y$ = 6.
Solution:
5x - 2y = 6
[Given equation of the line.]
- 2y = 6 - 5x
[Subtract 5x on both sides.]
y = 5 / 2x - 3
[Write in the form of y = mx + b.]
The equation y = 5 / 2x - 3 is in the form of y = mx + b, where slope m is 5 / 2 and intercept b is - 3.
The slope of perpendicular line is negative reciprocal of the slope of the line.
The slope of the perpendicular line is - 2 / 5.
The equation for the line in point - slope form of a line that passes through (6, - 5) with slope m = - 2 / 5 is
y + 5 = - 2 / 5(x - 6)
[Use (y - y_{1}) = m(x - x_{1}).]
So, the equation for the line in slope - point form that passes through (6, - 5), and perpendicular to 5x - 2y = 6 is y + 5 = - 2 / 5(x - 6).
Correct answer : (4)