# How do you write an equation of a line passing through (4, 2), perpendicular to y=2x+3?

Apr 18, 2017

See the entire solution process below:

#### Explanation:

The equation in the problem is in slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y = \textcolor{red}{2} x + \textcolor{b l u e}{3}$

Therefore the slope of this line is $m = 2$

Let us call the slope of the perpendicular line ${m}_{p}$. By definition, the slope of a perpendicular line is:

${m}_{p} = - \frac{1}{m}$

Therefore, for this problem, the slope of the perpendicular line is:

${m}_{p} = - \frac{1}{2}$

We can use the slope-intercept formula, substitute the values from the points for $x$ and $y$ and substitute the slope we determined and solve for $b$:

$2 = \left(\textcolor{red}{- \frac{1}{2}} \cdot 4\right) + \textcolor{b l u e}{b}$

$2 = - 2 + \textcolor{b l u e}{b}$

$\textcolor{red}{2} + 2 = \textcolor{red}{2} - 2 + \textcolor{b l u e}{b}$

$4 = 0 + \textcolor{b l u e}{b}$

$4 = \textcolor{b l u e}{b}$

Which means the equation is:

$y = \textcolor{red}{- \frac{1}{2}} x + \textcolor{b l u e}{4}$

Or, we could use the point-slope formula to also write and equation for the line. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the values from the point in the problem gives:

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- \frac{1}{2}} \left(x - \textcolor{red}{4}\right)$