Question

consider the line y=2/7x+5 find the equation of the line that is parallel and perpendicular to this line and passes through the point(-7,-5)

Answer 1

The equation of the line parallel to the given line and passing through (-7, -5) is y = ⅓x −4. The equation of the perpendicular line through the same point is y = -7/2x - 24.5.

Step-by-step explanation:

The student has asked to find the equation of a line that is parallel and perpendicular to the given line y = ⅓ x + 5 and that passes through the point (-7, -5).

The slope of the given line is ⅓, so any line parallel to this one will have the same slope. The equation of the parallel line through (-7, -5) is therefore y = ⅓ x + b. To find 'b', we use the point (-7, -5): -5 = (⅓)(-7) + b, which gives b = −4. The equation of the parallel line is y = ⅓ x −4.

To find the perpendicular line, we need a slope that is the negative reciprocal of ⅓, which is -7/2. The equation is y = -7/2 x + b. Using the point (-7, -5) again: -5 = (-7/2)(-7) + b, which gives b = -24.5. The equation of the perpendicular line is y = -7/2 x - 24.5.

Answer 2

Slope-intercept form is:

y = mx + b

"m" is the slope, "b" is the y-intercept (the y value when x = 0)

You need to find "m" and "b"

PARALLEL:

For lines to be parallel, they have to have the SAME slope. The given line's slope is 2/7, so the parallel line's slope is also 2/7

y = 2/7x + b

To find "b", you can plug in the point (-7, -5) into the equation

y = 2/7x + b

-5 = 2/7(-7) + b

-5 = -2 + b Add 2 on both sides

-3 = b

y = 2/7x - 3

PERPENDICULAR:

For lines to be perpendicular, their slopes have to be the opposite/negative reciprocal (flipped sign and number)

For example:

slope is 2

perpendicular line's slope is -1/2

slope is -2/5

perpendicular line's slope is 5/2

The given line's slope is 2/7, so the perpendicular line's slope is -7/2.

y = -7/2x + b

To find "b", plug in (-7,-5) into the equation

y = -7/2x + b

-5 = -7/2(-7) + b

-5 = 49/2 + b Subtract 49/2 on both sides

-5 - 49/2 = b Make the denominators the same

-10/2 - 49/2 = b

-59/2 = b

`y = -(7)/(2)x - (59)/(2)`