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

Mar 23, 2018

$y = 2 x - 1$

#### Explanation:

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

•color(white)(x)y=mx+b

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

$y = - \frac{1}{2} x + \frac{2}{3} \text{ is in this form}$

$\text{with slope m } = - \frac{1}{2}$

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

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

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

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

$\text{to find b substitute "(2,3)" into the partial equation}$

$3 = 4 + b \Rightarrow b = 3 - 4 = - 1$

$\Rightarrow y = 2 x - 1 \leftarrow \textcolor{red}{\text{equation of perpendicular line}}$

Mar 23, 2018

See below

#### Explanation:

If $y = m x + b$ is the line equation, we call Slope to value $m$ and y-intercept to value $b$

We know then that m´=-1/m is the slope of perpendiclar line to first. Thus we have

$m ' = - \frac{1}{- \frac{1}{2}} = 2$

Then, our perpendicular line is $y = 2 x + b$. Lets calculate b.

However $P \left(\textcolor{b l u e}{2} , \textcolor{red}{3}\right)$ is a point belonging to this new line, must be

color(red)3=2·color(blue)2+b, then $b = - 1$

The perpendicular line requested passing thru $P$ is $y = 2 x - 1$