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Equations of a Parallel and Perpendicular Line

Parallel and perpendicular line calculator

This calculator finds an equation for a parallel or perpendicular line passing through a given point. The calculator provides a step-by-step explanation for how to obtain the result.

Find the line that is parallel to $y = 0$ and passes though the point $(0, 0)$ .

solution

The equation of the line parallel to the given line that contains point $A$ is:

$y = 0$ ( General form )

$y = 0$ ( Slope y-intercept form )

explanation

Step 1:The slope of a given line is $m = 0$ .

Step 2: Parallel lines have the same slope, so the slope of the unknown line ( $m_{1}$ ) will also be $0$ . So the parallel line will have a slope of $m_{1} = 0$

Step 3: Now we have a point and the slope so we can use point-slope form, which is:

y - y_{0} = m_{1} (x - x_{0})

In this example we have: $m_{1} = 0$ , $x_{0} = 0$ and $y_{0} = 0$ . After substitution we have:

y - y_{0} = y - 0 = y = m_{1} (x - x_{0}) 0 (x - 0) 0

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Parallel and perpendicular lines calculator

input line in general or slope-intercept form
help ↓↓ examples ↓↓ tutorial ↓↓

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Input line in:

Line equation:

x

+

y

+

= 0

y = x

+

Input point:

( , )

Select what to compute:

Parallel line through the given point

Perpendicular line through the given point

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Examples

ex 1:

Find the equation of the line that is parallel to

2 x + y - 2 = 0

and passes though the point

(3, 1)

.

ex 2:

Find the equation of the line that is perpendicular to

y = 2 x - 5

and passes though the point

(- \frac{2}{3}, - \frac{1}{4})

.

Related calculators

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2:

3:

Line - point distance

4:

Lines intersection

5:

Find more worked-out examples in our database of solved problems.

How to find line through a point parallel to a given line ?

Equation of the line that passes through the point $A (x_{0}, y_{0})$ and is parallel to the line $y = m x + b$ is:

y - y_{0} = m (x - x_{0})

Example:

parallel lines

Find the equation of the line that passes through the point $A (- 1, 2)$ and is parallel to the line $y = 2 x - 3$

Solution:

In this example we have: $x_{0} = - 1, y_{0} = 2, m = 2$ . So we have:

y - y_{0} y - 2 y - 2 y y = m (x - x_{0}) = 2 (x - (- 1)) = 2 x + 2 = 2 x + 2 + 2 = 2 x + 4

Note : If you want to solve this problem using above calculator, you need to rewrite line equation in general form ( $2 x - y - 3 = 0$ )

How to find line through a point perpendicular to a given line ??

Equation of the line that passes through the point $A (x_{0}, y_{0})$ and is perpendicular to the line $y = m x + b$ is:

y - y_{0} = - \frac{1}{m} (x - x_{0})

Example:

Find the equation of the line that passes through the point $A (- 1, 2)$ and is perpendicular to the line $y = 2 x - 3$

Solution:

perpendicular lines

In this example we have: $x_{0} = - 1, y_{0} = 2, m = 2$ . So we have:

y - y_{0} y - 2 y - 2 (y - 2) \cdot 2 2 (y - 2) 2 y - 4 2 y y = - \frac{1}{m} (x - x_{0}) = - \frac{1}{2} (x - (- 1)) = - \frac{1}{2} (x + 1) = - \frac{1}{2} \cdot 2 (x + 1) = - (x + 1) = - x - 1 = - x + 3 = - \frac{1}{2} x + \frac{3}{2}

Note : If you want to solve this problem using above calculator, you need to rewrite line equation in general form ( $2 x - y - 3 = 0$ )

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