Writing equations of lines parallel and perpendicular to a given line through a point

Question

Consider the line y = -7x + 9

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

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

Answer 1

A parallel line will have the same slope but different y intercept. So for the first one, you know the slope will be -7. Additionally, you know that it passes through (3, -3). Therefore, you can write the following: -3 = -7(3) +b, and solving for b, you get 18. Therefore, the equation of your line will be y = -7x + 18.

A perpendicular line's slope will be opposite reciprocal to the original line. This means to reverse the sign and to divide the number by one. So since the slope of the original line is -7, the slope of the new line will be 1/7. From there, you can plug in (3, -3) to find the y-intercept, or b: -3 = 3(1/7) + b, so b = -24/7. Therefore, the equation of this line is y = (1/7)x - 24/7.

Let me know if you have any additional questions!

Answer 2

Hi Katie,

Thanks for your question! Given any line of slope m, the line perpendicular to it has slope -1/m. In this case, our line is

y = -7x + 9,

so m = -7. Therefore, the line perpendicular to this line has slope -(1/(-7)), or simply 1/7. Now we know the slope of the perpendicular line, and we know that this line passes through the point (3,-3). Since we are given a point and a slope, we can use the point-slope formula to get the equation of a line:

y - y0 = m(x - x0).

Here, m is the slope of the perpendicular line (m = 1/7) and (x0,y0) is the point that this line passes through (so x0 = 3 and y0 = -3). Plugging these values into the point-slope formula, we get

y - (-3) = 1/7(x - 3)

y + 3 = 1/7*x -3/7

y = 1/7*x - 3/7 - 3

y = 1/7*x - 24/7,

which is the final answer in slope-intercept form.

Answer 3

A) For an equation to be parallel, you need to have the same slope as the given equation.

For the given equation y = -7x + 9, the slope is -7 so the equation that is parallel has to also have a slope of -7.

y = mx + b

y = -7x + b Now, we plug in the given point (3, -3)

for x and y and solve for b

-3 = -7(3) + b

-3 = -21 + b

+21 +21

17 = b

So, now that we found b, we plug it into our y = mx + b.

Hence, the equation parallel to y = -7x + 9 that passes through the point (3, -3) is y = -7x + 17.

B) For an equation to be perpendicular, the slope of the new equation must be an opposite reciprocal of the original slope.

For the given equation y = -7x + 9, the slope is -7 so the opposite reciprocal 1/7.

y = mx + b

y = 1/7x + b

-3 = 1/7(3) + b Plug in the given point (3, -3)

for x and y and solve for b

-3 = 3/7 + b

-3/7 -3/7

-24/7 = b

So, now that we found b, we plug it into our y = mx + b.

Hence, the equation perpendicular to y = -7x + 9 that passes through the point (3, -3) is y = 1/7x - 24/7.