Question

Consider the line y=8x-7

Find the equation of the line that is parallel to this line and passes through the point (5,-3)

Find the equation of the line that is perpendicular to this line and passes through the point (5,-3)

Answer 1

Parallels lines:
y = 8x - 43

Perpendicular line:

y =
`(-1)/(8)`x -
`(19)/(8)`

Explanation:

y = 8x -7

Parallel lines have the same slope.

The slope will be 8. We will use the x from the point (5,-3) and the y from the point (5,-3) to find the y-intercept (b)

y = mx + b Substitute in -3 for y, 8 for m, and 5 for x.

-3 = 8(5) + b

-3 = 40 + b Subtract 40 from both sides

-3 - 40 = 40 - 40 + b

-43 = b

Substitute in 8 for m and -43 for b to write the equation

y = mx + b

y = 8x - 43

Perpendicular slope are opposite reciprocals of each other, so the perpendicular slope is
`(-1)/(8)`

Substitute
`(-1)/(8)` doe m, -3 for y and 5 for x.

y = mx + b

-3 =
`(-1)/(8)`(5) + b

-3 =
`(-5)/(8)` + b add
`(5)/(8)` from both sides

-3 +
`(5)/(8)` =
`(-5)/(8)` +
`(5)/(8)` + b

`(-24)/(8)` +
`(5)/(8)` = b

`(-19)/(8)` = b

Substitute
`(-1)/(8)` for m and
`(-19)/(8)` for b

y = mx + b

y =
`(-1)/(8)`x -
`(19)/(8)`

Helping in the name of Jesus.

Answer 2

Since the line we are looking for is parallel to the given line, it has the same slope. Therefore, the slope of the line we are looking for is 8. Also, we know that the line passes through the point (5,-3). Using the point-slope form of a line, we can write the equation of the line as:

y - (-3) = 8(x - 5)

Simplifying this equation, we get:

y = 8x - 43

Therefore, the equation of the line that is parallel to y = 8x - 7 and passes through the point (5,-3) is y = 8x - 43.

Since the line we are looking for is perpendicular to the given line, its slope is the negative reciprocal of the slope of the given line. The slope of the given line is 8, so the slope of the line we are looking for is -1/8. Also, we know that the line passes through the point (5,-3). Using the point-slope form of a line, we can write the equation of the line as:

y - (-3) = (-1/8)(x - 5)

Simplifying this equation, we get:

y = (-1/8)x + (41/8)

Therefore, the equation of the line that is perpendicular to y = 8x - 7 and passes through the point (5,-3) is y = (-1/8)x + (41/8).