Question

Write an equation of the line that passes through the given point and is (a) parallel and (b) perpendicular to the given line.

Answer 1

Answer:

$\textsf{a)} \quad y=-4x+19$

$\textsf{b)} \quad y=\dfrac{1}{4}x+2$

Step-by-step explanation:

From inspection of the graph, two points on the line are:

• (1, 6)
• (2, 2)

Substitute these points in the slope formula to find the slope of the line:

$\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{2-6}{2-1}=\dfrac{-4}{1}=-4$

Part (a)

Lines that are parallel have the same slope.

Therefore, a line parallel to the given line has a slope of -4.

To write an equation of the parallel line, substitute the found slope and point (4, 3) into the point-slope formula and rearrange:

$\implies y-y_1=m(x-x_1)$

$\implies y-3=-4(x-4)$

$\implies y-3=-4x+16$

$\implies y=-4x+19$

Part (b)

If two lines are perpendicular to each other, their slopes are negative reciprocals.

Therefore, a line perpendicular to the given line has a slope of ¹/₄.

To write an equation of the parallel line, substitute the found slope and point (4, 3) into the point-slope formula and rearrange:

$\implies y-y_1=m(x-x_1)$

$\implies y-3=\dfrac{1}{4}(x-4)$

$\implies y-3=\dfrac{1}{4}x-1$

$\implies y=\dfrac{1}{4}x+2$