How do you write the equation in point slope form given point (5,8) and perpendicular to the line passing though the points (-8,7) and (-2,0)?

Jul 21, 2017

See a solution process below:

Explanation:

First, we need to find the slope of the line passing through the two points in the problem. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{0} - \textcolor{b l u e}{7}}{\textcolor{red}{- 2} - \textcolor{b l u e}{- 8}} = \frac{\textcolor{red}{0} - \textcolor{b l u e}{7}}{\textcolor{red}{- 2} + \textcolor{b l u e}{8}} = - \frac{7}{6}$

The formula for a perpendicular slope $\left({m}_{p}\right)$ is:

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

Substituting the slope we calculated gives:

${m}_{p} = - \frac{1}{- \frac{7}{6}} \implies \frac{6}{7}$

We can now use the point-slope formula to write an equation for the line from the problem. 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 $\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ is a point the line passes through.

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

$\left(y - \textcolor{red}{8}\right) = \textcolor{b l u e}{\frac{6}{7}} \left(x - \textcolor{red}{5}\right)$

If necessary, we can solve for $y$ to transform this equation to 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.

$y - \textcolor{red}{8} = \left(\textcolor{b l u e}{\frac{6}{7}} \times x\right) - \left(\textcolor{b l u e}{\frac{6}{7}} \times \textcolor{red}{5}\right)$

$y - \textcolor{red}{8} = \frac{6}{7} x - \frac{30}{7}$

$y - \textcolor{red}{8} + 8 = \frac{6}{7} x - \frac{30}{7} + 8$

$y - 0 = \frac{6}{7} x - \frac{30}{7} + \left(\frac{7}{7} \times 8\right)$

$y = \frac{6}{7} x - \frac{30}{7} + \frac{56}{7}$

$y = \frac{6}{7} x + \frac{- 30 + 56}{7}$

$y = \textcolor{red}{\frac{6}{7}} x + \textcolor{b l u e}{\frac{26}{7}}$