How to find a perpendicular line that passes through a point

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Find parallel and perpendicular lines step by step
3 Answers By Expert Tutors
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What is the formula for perpendicular lines?

Find parallel and perpendicular lines step by step

The calculator will find the equation of the parallel/perpendicular line to the given line passing through the given point, with steps shown.

For drawing lines, use the graphing calculator.

Solution

Your input: find the equation of the line parallel to the line $$$y=2 x + 5$$$ passing through the point $$$\left(-3,5\right)$$$.

The equation of the line in the slope-intercept form is $$$y=2 x + 5$$$.

The slope of the parallel line is the same: $$$m=2$$$.

So, the equation of the parallel line is $$$y=2 x+a$$$.

To find $$$a$$$, we use the fact that the line should pass through the given point: $$$5=\left(2\right) \cdot \left(-3\right)+a$$$.

Thus, $$$a=11$$$.

Therefore, the equation of the line is $$$y=2 x + 11$$$.

Answer: $$$y=2 x + 11$$$.

When you multiply the slopes of perpendicular lines the product is always -1. Therefore, we must first find the slope of the given line. To do this we need to put the equation given into the form of y=mx+b, where m is the slope and b is the y-intercept.

3x-6y=-11 original equation

3x=-11+6y add 6y to both sides

3x+11=6y add 11 to both sides to isolate the "y" term

6y=3x+11 rearrange

y=(3x/6)+(11/6) divide both sides by 6

y=(1/2)x+(11/6) simplify

From here we can see that the slope of the original line is 1/2. The slope of the new line has to be the negative inverse of this in order to make the products of the slopes equal to -1. Therefore, the slope of the new line is -2. Using this for m and the point (4,-5) for x and y, we can solve for b:

y=mx+b

-5=(-2)(4)+b

-5=-8+b

b=8-5

b=3

With m = -2 and b = 3, we now get for y=mx+b--->y=-2x+3

Hope this helps!

Bachiko Z.

asked • 10/18/19

3 Answers By Expert Tutors

Roderick W. answered • 10/20/19

Certified Math Teacher Looking To Help!

Rewrite the equation 8x + 7y − 4 = 0 in slope-intercept (y = mx+b) form.

Add 4 to both sides ---------------> 8x + 7y = 4

Subtract 8x from both sides ----> 7y = -8x + 4

Divide by 7 on both sides -------> y = -8/7x + 4/7

So the original line has a slope of -8/7.

The line perpendicular to this will have the slope 7/8 because perpendicular lines have opposite reciprocal slopes (change the sign, flip the fraction).

The line that has a slope of 7/8 and passes through the point (2,4) can be found by substituting the slope (7/8) and the point (2,4) in the equation y = mx + b.

Let y = 4 (because that's the y-coordinate of the point).

Let m = 7/8 (because m is the slope).

Let x = 2 (because that's the x-coordinate of the point).

You will be solving for b, the y-intercept of your line.

Substitute values in for y, m, and x ----> (4) = (7/8)(2) + b

Multiply 7/8 and 2 --------------------------> 4 = 14/8 + b

Simplify 14/8 --------------------------------> 4 = 7/4 + b

Subtract 7/4 from both sides -----------> 9/4 = b

The y-intercept (b) of the new line is 9/4. The slope (m) is 7/8. Put this in slope-intercept form (y = mx + b).

y = 7/8x + 9/4

If standard form (Ax + By = C) is required, then rewrite this in standard from by subtracting 7/8x from both sides.

-7/8x + y = 9/4

Then multiply both sides of the equation by -8 to get an integer coefficient for standard form.

7x - 8y = -18

y = -(8/7)x + 4/7

Is 8x + 7y - 4 = 0 in Slope Intercept Form

y = (7/8)x + 9/4 is the Perpendicular Line in the Slope Intercept Form

This is the perpendicular line is set equal to zero below

-(7/2)x + 4y - 9 = 0

To find the line perpendicular to

8x + 7y - 4 = 0

First put the line above in the Slope Intercept Form by solving for y

8x + 7y -4 = 0

Solve for y

7y = -8x + 4

y = (-8/7)x + 4/7

In the Slope Intercept Form, the coefficient of x is the slope m

y = (-8/7)x + 4/7

y = mx + b

m = -8/7

Look at the slope of the line of above

m = -8/7

The slope of the perpendicular line is the inverse of the slope above with the opposite sign

7/8 is the inverse of -8/7 with the opposite sign

m = 7/8 for the Perpendicular line

Write the Perpendicular Line in the Slope Intercept form

y = mx + b

y = 7/8 + b

Use the coordinates (2, 4) in the Slope Intercept form of the Perpendicular line to solve for b

4 = (7/8) * 2 + b

4 = 7/4 + b

Subtract 7/4 for both sides of the equation to solve for b

4 - 7/4 = b

16/4 - 7/4 = b

9/4 = b

Then write the completed Perpendicular line in the Slope Intercept Form

y = (7/8)x + 9/4

You can set this line equal to zero to yield

-(7/8)x + y = 9/4

-(7/2)x + 4y = 9

-(7/2)x + 4y - 9 = 0

You can graph the lines to check that they are perpendicular.

It is easier to graph the lines in Slope Intercept Form, you can plug them in at Desmos.com

I hope you find this useful if you have any questions send me a message.

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What is the formula for perpendicular lines?

Consider the equation of the line is ax + by + c = 0 and coordinates are (x₁, y₁), the slope should be − a/b. If one line is perpendicular to this line, the product of slopes should be -1. Let m₁ and m₂ be the slopes of two lines, and if they are perpendicular to each other, then their product will be -1.

find perpendicular line passes point Perpendicular lines