Step: 1

The slope-intercept form of the equation of a line with slope m and y -intercept b is y = m x + b .

Step: 2

Slope of the line y = - 2x + 4 is - 2.

[Compare with y = m x + 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

[Solve for b .]

Step: 8

The y -intercept is b = 8.

Step: 9

[Replace m with - 2 and b with 8 in y = mx + b .]

Step: 10

Correct Answer is : y = - 2x + 8

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 = - 3 5 x + 5 is - 3 5

[Compare with the slope intercept form equation.]

Step: 3

The slope of line parallel to y = - 3 5 x + 5 is - 3 5 .

[Parallel lines have equal slopes.]

Step: 4

The equation of the line with slope - 3 5 in slope-intercept form is y = - 3 5 x + b

Step: 5

Substitute the point (- 2, 6) in the above equation.

Step: 6

6 = - 3 5 (- 2) + b

[Replace x with - 2 and y with 6]

Step: 7

[Solve for b .]

Step: 8

[Replace m with - 3 5 and b with 24 5 .]

Step: 9

The equation of the line parallel to y = - 3 5 x + 5 is y = - 3 5 x + 24 5 which is passing through the point (- 2, 6).

Correct Answer is : y = - 3 5 x + 24 5

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

[Solve for b .]

Step: 8

The y -intercept is b = 10.

Step: 9

[Replace m with 4 and b with 10 in y = mx + b .]

Step: 10

Correct Answer is : y = 4x + 10

Step: 1

The slope-intercept form of the equation of a line with slope m and y -intercept b is y = m x + 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 - 1 4 .

[Product of slopes of perpendicular lines is - 1.]

Step: 4

Step: 5

[Substitute (x _{1}, y _{1}) = (- 2, - 4) and m = - 1 4 in the equation in step 4.]

Step: 6

[Simplify the equation.]

Step: 7

The equation of the line passing through the point (- 2, - 4) is y + 4 = - 1 4 (x + 2).

Correct Answer is : y + 4 = - 1 4 ( x + 2 )

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 - 1 3 .

[Product of slopes of perpendicular lines is - 1.]

Step: 4

Step: 5

Point (x _{1}, y _{1}) = (2, 3) and slope m of the perpendicular line = - 1 3 .

Step: 6

[Substitute x _{1} = 2, y _{1} = 3 and m = - 1 3 in the equation in step 4.]

Step: 7

The equation of the line passing through the point (2, 3) is y - 3 = - 1 3 (x - 2).

Correct Answer is : y - 3 = - 1 3 (x - 2)

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 - 1 3 .

[Product of slopes of perpendicular lines is - 1.]

Step: 4

Step: 5

Point (x _{1}, y _{1}) = (- 3, 5) and slope m of the perpendicular line = - 1 3

Step: 6

[Substitute x _{1} = - 3, y _{1} = 5 and m = - 1 3 in the equation in step 4.]

Step: 7

[Simplify the equation.]

Step: 8

So, the equation of the line passing through the point (- 3, 5) is y - 5 = - 1 3 (x + 3).

Correct Answer is : y - 5 = - 1 3 (x + 3)

Step: 1

The slope of the line represented by x + 4y - 9 = 0 is - 1 4

[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 = - 1 4

[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 - y _{1} = m (x - x _{1})

[Use slope - point form of a line.]

Step: 5

Step: 6

4y - 24 = - x + 5

Step: 7

Correct Answer is : x + 4y - 29 = 0

Step: 1

The slope of the line represented by x + 4y - 24 = 0 is - 1 4 .

[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 = - 1 4

[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 - y _{1} = m (x - x _{1})

[Use the slope - point form of line.]

Step: 5

Step: 6

4y - 20 = - x + 2

Step: 7

Correct Answer is : x + 4y - 22 = 0

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 = - 4 7 x + 7 is - 4 7

[Compare with the slope intercept form equation.]

Step: 3

The slope of line parallel to y = - 4 7 x + 7 is - 4 7 .

[Parallel lines have equal slopes.]

Step: 4

The equation of the line with slope - 4 7 in slope-intercept form is y = - 4 7 x + b

Step: 5

Substitute the point (- 3, 6) in the above equation.

Step: 6

6 = - 4 7 (- 3) + b

[Replace x with - 3 and y with 6]

Step: 7

[Solve for b .]

Step: 8

[Replace m with - 4 7 and b with 30 7 .]

Step: 9

The equation of the line parallel to y = - 4 7 x + 7 is y = - 4 7 x + 30 7 which is passing through the point (- 3, 6).

Correct Answer is : y = - 4 7 x + 30 7

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

[Solve for b .]

Step: 8

The y -intercept is b = 9

Step: 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

Step: 1

The given equation is 2y = - 3x + 6.

Step: 2

⇒ y = - 3 2 x + 3

[Writing the given equation in slope-intercept form y = mx + b .]

Step: 3

Slope of the line y = - 3 2 x + 3 is - 3 2 .

[Where m is the slope.]

Step: 4

So, slope of the line which is perpendicular to the line 2y = - 3x + 6 is 2 3 .

[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 (x _{1}, y _{1}) with slope m in point-slope form is y - y _{1} = m (x - x _{1}).

Step: 7

⇒ y - 5 = 2 3 (x - 1)

[Substitute x _{1} = 1 and y _{1} = 5 and m = 2 3 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 = 2 3 (x - 1) .

Correct Answer is : y - 5 = 2 3 (x - 1)

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 = 1 3 .

[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 (x _{1}, y _{1}) with slope ′m ′ in point-slope form is y - y _{1} = m (x - x _{1}).

Step: 5

Point (x _{1}, y _{1}) = (- 4, - 4) and slope m of the perpendicular line = - 3.

Step: 6

[Substitute x _{1} = - 4, y _{1} = - 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)

Step: 1

The slope-intercept form of the equation of a line with slope m and y -intercept b is y = m x + 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 - 1 8 .

[Product of slopes of perpendicular lines is - 1.]

Step: 4

The equation of the line passing through the point (x _{1}, y _{1}) with slope m in point-slope form is y - y _{1} = m (x - x _{1}).

Step: 5

[Substitute (x _{1}, y _{1}) = (4, 6) and m = - 1 8 in the equation in step 4.]

Step: 6

[Simplify the equation.]

Step: 7

Step: 8

Step: 9

The equation of the line passing through the point (4, 6) is y = - 1 8 (x ) + 13 2

Correct Answer is : y = - 1 8 (x ) + 13 2

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

[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 (x _{1}, y _{1}) with slope m is y - y _{1} = m (x - x _{1}).

[Point-slope form.]

Step: 7

[Substitute x _{1} = 4, y _{1} = 6 and m = - 4 in the equation in step 6.]

Step: 8

Step: 9

Step: 10

The equation of the line in point-slope form is y = - 4x + 22

Correct Answer is : y = - 4x + 22

Step: 1

The slope-intercept form of the equation of a line with slope m and y -intercept b is y = m x + b .

Step: 2

The equation is y = 4 5 x + 6.

Step: 3

Comparing the equation with equation in step 1, the slope of the line is 4 5 .

Step: 4

So, the slope of line parallel to the line is 4 5 .

[Slope of parallel lines are same.]

Step: 5

The parallel line passes through the point (x _{1}, y _{1}) = (3, 5) and has a slope m = 4 5 .

Step: 6

The equation of the line passing through the point (x _{1}, y _{1}) with slope m in point-slope form is y - y _{1} = m (x - x _{1}).

Step: 7

[Substitute x _{1} = 3, y _{1} = 5 and m = 4 5 in the equation in step 6.]

Step: 8

Step: 9

Step: 10

The equation of the parallel line in point-slope form is y = 4 5 (x ) + 13 5

Correct Answer is : y = 4 5 (x ) + 13 5

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