# How do you find equation of line which is perpendicular to the line 2x-3y=5 and passes through the point (2,1)?

Jan 3, 2017

Swap the coefficients, make one negative, and then use the point to solve for the new constant.

#### Explanation:

When given the equation of a line in the form:

$a x + b y = c$

It is very easy to make a line that is perpendicular to the given line:

1. Swap a and b
2. Change the sign of either a or b
3. Make the right side an arbitrary constant, k

The two possible results are shown below:

$b x - a y = k \text{ [1]}$

or

$- b x + a y = k \text{ [2]}$

For the given line, $2 x - 3 y = 5$, $a = 2 \mathmr{and} b = - 3$, therefore, we shall use form of equation [2], because it makes both constants positive:

$3 x + 2 y = k \text{ [3]}$

Given the point $\left(2 , 1\right)$, we can find the value of k by substituting 2 for x and 1 for y into equation [3]:

$3 \left(2\right) + 2 \left(1\right) = k \text{ [3]}$

$k = 7$

Substitute 7 for k into equation [3]:

$3 x + 2 y = 7 \text{ [4]}$