How do you write an equation of the line in slope-intercept form given the slope and a point that lies on the line: m=-1/13 and (-7,5)?

Dec 26, 2016

The slope-intercept for of the equation meeting the requirements of this problem is:

$y = \textcolor{b l u e}{- \frac{1}{13}} x + \textcolor{red}{\frac{58}{13}}$

Explanation:

To first identify the equation we can use the point-slope formula and then translate into the slope-intercept form.

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.

We have been given the slope $\textcolor{b l u e}{m = - \frac{1}{13}}$

We have been given a point on the line $\textcolor{red}{\left(\left(- 7 , 5\right)\right)}$

Substituting gives:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- \frac{1}{13}} \left(x - \textcolor{red}{- 7}\right)$

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- \frac{1}{13}} \left(x + \textcolor{red}{7}\right)$

The slope-intercept form of a linear equation is:

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

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

We can solve for $y$ to put our equation into this form:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- \frac{1}{13}} \left(x + \textcolor{red}{7}\right)$

(y - color(red)(5)) = color(blue)(-1/13)x + (color(blue)(-1/13) * color(red)(7)))

$y - \textcolor{red}{5} = \textcolor{b l u e}{- \frac{1}{13}} x + \left(\textcolor{b l u e}{- \frac{7}{13}}\right)$

$y - 5 = \textcolor{b l u e}{- \frac{1}{13}} x - \frac{7}{13}$

$y - 5 + \textcolor{red}{5} = \textcolor{b l u e}{- \frac{1}{13}} x - \frac{7}{13} + \textcolor{red}{5}$

$y - 0 = \textcolor{b l u e}{- \frac{1}{13}} x - \frac{7}{13} + \left(\textcolor{red}{5} \cdot \frac{13}{13}\right)$

$y = \textcolor{b l u e}{- \frac{1}{13}} x - \frac{7}{13} + \textcolor{red}{\frac{65}{13}}$

$y = \textcolor{b l u e}{- \frac{1}{13}} x + \textcolor{red}{\frac{58}{13}}$