# How do you write an equation that is perpendicular to y = -1/2x + 2/3, passes through (2,3)?

Nov 13, 2016

$y = 2 x - 1$

#### Explanation:

Given:$\text{ } y = - \frac{1}{2} x + \frac{2}{3}$

Compare to $y = m x + c$ where $m$ is the gradient (slope)

A straight line graph that is perpendicular to this would have the gradient of $- \frac{1}{m}$. So in this case $- \frac{1}{m} = + \frac{2}{1} = 2$

So we know that the line we are looking for is of form:

$y = 2 x + c$

This has 3 unknowns so we need to change it so that there is only 1 unknown and thus solvable.

We are told that it passes through the point $\left(x , y\right) \to \textcolor{red}{\text{(2,3)}}$ thus we have some known values. Lets substitute these:

$\textcolor{g r e e n}{y = 2 x + c \text{ "->" } \textcolor{red}{3} = 2 \left(\textcolor{red}{2}\right) + c}$

$3 = 4 + c$

$c = - 1$

So we now have $y = 2 x - 1$