How do you find the point-slope form of the equation of the line passing through the points (-7, 0) and (5, 4)?

Jun 14, 2018

$y - 4 = \frac{1}{3} \left(x - 5\right)$

Explanation:

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

•color(white)(x)y-b=m(x-a)

$\text{where m is the slope and "(a,b)" a point on the line}$

$\text{to calculate m use the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(-7,0)" and } \left({x}_{2} , {y}_{2}\right) = \left(5 , 4\right)$

$m = \frac{4 - 0}{5 - \left(- 7\right)} = \frac{4}{12} = \frac{1}{3}$

$\text{use either of the 2 given points as point on the line}$

$\text{using "(5,4)" then}$

$y - 4 = \frac{1}{3} \left(x - 5\right) \leftarrow \textcolor{red}{\text{in point-slope form}}$

Jun 14, 2018

$\left(y - 4\right) = \frac{1}{3} \left(x - 5\right)$

Explanation:

First you determine the slope:

$\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right) = \left(- 7 , 0\right)$

$\left(\textcolor{red}{{x}_{2}} , \textcolor{red}{{y}_{2}}\right) = \left(5 , 4\right)$

$\textcolor{g r e e n}{m} = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

$\textcolor{g r e e n}{m} = \frac{\textcolor{red}{4} - \textcolor{b l u e}{0}}{\textcolor{red}{5} - \textcolor{b l u e}{\left(- 7\right)}} = \frac{4}{12} = \frac{1}{3}$

Now use the Point Slope form of a line:

You can use any point on the line, let's use the second one since both values are positive:

$\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{g r e e n}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

$\left(y - \textcolor{b l u e}{4}\right) = \textcolor{g r e e n}{\frac{1}{3}} \left(x - \textcolor{b l u e}{5}\right)$