# How do you write an equation of a line passing through (2, 4), perpendicular to 10x - 5y = 8?

Jan 8, 2018

$y = - \frac{1}{2} x + 5$

#### Explanation:

First, convert $10 x - 5 y = 8$ into the form of $y = m x + b$

$10 x - 5 y = 8$

$- 5 y = 8 - 10 x$

$y = 2 x - \frac{8}{5}$

The perpendicular line to $y = 2 x - \frac{8}{5}$ will have a slope equal to $- \frac{1}{2}$, because perpendicular lines' slopes have a product of $- 1$.

The new line will be

$y = - \frac{1}{2} x + b$

We know that it passes through $\left(2 , 4\right)$, so we can plug that into the new equation.

$4 = - \frac{1}{2} \cdot 2 + b$

$4 = - 2 + b$

$b = 6$

So, the new equation of the line will be

$y = - \frac{1}{2} x + 5$

Jan 8, 2018

$y = - \frac{1}{2} x + 5$

#### Explanation:

$\text{given a line with slope m then the slope of a line }$
$\text{perpendicular to it is}$

•color(white)(x)m_(color(red)"perpendicular")=-1/m

$\text{the equation of a line in "color(blue)"slope-intercept form}$ is.

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{rearrange "10x-5y=8" into this form}$

$\Rightarrow - 5 y = - 10 x + 8 \Rightarrow y = 2 x - \frac{8}{5} \Rightarrow m = 2$

$\Rightarrow {m}_{\textcolor{red}{\text{perpendicular}}} = - \frac{1}{2}$

$\Rightarrow y = - \frac{1}{2} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute "(2,4)" into the partial equation}$

$4 = - 1 + b \Rightarrow b = 4 + 1 = 5$

$\Rightarrow y = - \frac{1}{2} x + 5 \leftarrow \textcolor{red}{\text{perpendicular equation}}$