Main content

### Course: High school geometry > Unit 6

Lesson 5: Equations of parallel & perpendicular lines- Parallel lines from equation
- Parallel lines from equation (example 2)
- Parallel lines from equation (example 3)
- Perpendicular lines from equation
- Parallel & perpendicular lines from equation
- Writing equations of perpendicular lines
- Writing equations of perpendicular lines (example 2)
- Write equations of parallel & perpendicular lines
- Proof: parallel lines have the same slope
- Proof: perpendicular lines have opposite reciprocal slopes
- Analytic geometry FAQ

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Writing equations of perpendicular lines (example 2)

Sal finds the equation of a line perpendicular to a line given in slope-intercept form that passes through a specific point.

## Want to join the conversation?

- why did you substract values of point from y and x in equation?

can you explain, how it works?(12 votes)- It's because he used "point-slope form", which is always in the form of y-a=m(x-b), see also https://www.khanacademy.org/math/algebra/two-var-linear-equations/point-slope/v/idea-behind-point-slope-form.(13 votes)

- I do not understand exactly what I should do . My teacher told me that we would flip like you said -2/5 to 5/2 . I don't understand why you have to flip the fraction.(6 votes)
- The inverse of -2/5 is 5/2, as you know. Perpendicular lines have inverse slopes.(8 votes)

- Do you HAVE to use point slope form here?(5 votes)
- No, you don't have to use point slope.

You could use the slope intercept form, y = mx +b. We know from the video that m=5/2. You can figure out what b is by setting x=2 and y=8.(7 votes)

- Couldn't you just substitute the points given into the equation, then you would reciprocate and change the sign of the slope..... correct me if I'm wrong cause point slope just seems alot more confusing(3 votes)
- No because the point is not necessarily on the original line. If you assume they are on the original line, then you would get the wrong value for y intercept.

Lets try: say y = 2x + 3 and you want a perpendicular that goes through (-2,4)

Rather than point slope, I will use the slope intercept equation and substitute

y = =1/2 x + b, so 4 = -1/2 (-2) + b, 4 = 1 + b, b = 3 and your perpendicular equation is y = -1/2 x + 3

If I tried to do your suggestion of just substituting, I would get 4 = 2 (-2) + 3 so 4=1 which does not make sense If you substituted looking for b 4 = 2 (-2) + b, so b = 8 which is different from the correct equation above.(7 votes)

- How would you write and equation for a perpendicular line x= -3, through (2,1)?(3 votes)
- x= lines are vertical, so a perpendicular line would be a horizontal line in the form y = #. Since the y intercept of the equation must be 1, the answer is y = 1.(3 votes)

- What is point-slope form?(3 votes)
- If we have a point on the line with coordinates (y₁, x₁), and the line has a slope of m, the point-slope form of the line is y-y₁=m(x-x₁)(3 votes)

- What is the meaning of that : "Lines with undefined slopes have the form of x = c, where c is the x-intercept."?(3 votes)
- If a line has an undefined slope, that's because the "change in x" that is the denominator of the slope equals zero.

Since there's no "change in x" (but y can take on any value), the line is vertical.

In order to write its equation, all we need to know is where it crosses the X-axis and we call that point c, giving us the equation x = c.

So, for example, x = -4 would be a vertical line crossing the X-axis at -4

and x = 0 would be a vertical line coinciding with the Y-axis.

Hope this helps.(3 votes)

- at7:39, what is the slope of the line perpendicular to 2x+y=5(2 votes)
- Solving for y, you get y = -2x + 5, so slope of this line is -2, the slope of the perpendicular line would be the opposite reciprocal, opposite sign of -2 is positive, and reciprocal of 2 is easier to see if you divide by 1 to get 2/1, thus reciprocal is 1/2 which is also the answer to your question. If you wanted it in standard form, you switch coefficients of x and y and change sign of y to get x - 2y = #.(3 votes)

- why did you substract values of point from y and x in equation?

can you explain, how it works?(1 vote)- Sal was taking the slope and a point on the line and using that to create the line's equation in point-slope form. Point-slope form is y-y1 = m(x-x1), where x1 and y1 are the values of the known point on the line, and m is the slope. Sal was substituting in the values that he had in that equation to get the line's equation.

More information about point-slope form can be found at this Khan Academy link: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:point-slope-form/v/idea-behind-point-slope-form(3 votes)

- what if they are paraallel(1 vote)
- Then instead of the second line's slope being the negative reciprocal of the first line's, the slopes would be the same. Everything else would be the same.(3 votes)

## Video transcript

- [Instructor] Find the
equation of a line perpendicular to this line that passes to
the point two comma eight. So this first piece of information that it's perpendicular to that
line right over there. What does that tell us? Well if it's perpendicular to this line, it's slope has to be the
negative inverse of two-fifths. So its slope, the negative
inverse of two-fifths, the inverse of two-fifths is five. Let me do it in a better color. A nicer green. If this lines slope is
negative two-fifths, the equation of the line
we have to figure out that's perpendicular, the slope is going to be the inverse. So instead of two-fifths, it's gonna be five halves. And instead of being a negative, it's going to be a positive. So this is the negative inverse of negative two-fifths, right. You take the negative
sign, it becomes positive. You swap the five and the two, you get five halves. So that is going to have to be our slope. And we can actually use the
point slope form right here. It goes through this point right there. So let's use point slope form. Y minus this Y value which
has to be on the line. Is equal to our slope, five halves times X minus this X value. The X value when Y is equal to eight. And this is the equation of
the line in point slope form if you wanna put it in slope intercept form. You can just do a little bit of algebra. Algebraic manipulation. Y minus eight is equal to let's
distribute the five halves. So five halves X minus five
halves times two is just five. And then add eight to both sides. You get Y is equal to five halves X. Add eight to negative five. So plus three. And we are done.