91Ó°ÊÓ

The slope of a line is the ratio of the change in the y-coordinates to the change in the x- coordinates.

The slope-intercept form of the equation of a line is given by $y=mx+b$ where mis the slope and bis the y-intercept.

The point slope form of a equation of a line is given by $y-y_1=m(x-x_1)$where $\left(x_1,y_1\right)$ are the coordinates of a point on the line and m is the slope of the line.

Given line is $x+3y=7$.

In order to convert it into the slope intercept form, keep the variable y on the left hand side and bring rest of the things to the right hand side.

Therefore,

$\begin{array}{c}x+3y=7\\ ⇒x+3y−x=1−x(\text{subtract}x\text{frombothsides)}\\ ⇒3y=7−x\end{array}$

Divide both sides by 3,

$\begin{array}{c}⇒\frac{3y}{3}=\frac{7−x}{3}\\ ⇒y=\frac{7}{3}−\frac{x}{3}\\ ⇒y=−\frac{1}{3}x+\frac{7}{3}\end{array}$

This is of the form $y=mx+b$

Hence, the slope is $m=\frac{-1}{3}$

Given the point $\left(5,2\right)$ and slope is $m=-\frac{1}{3}$.

Therefore by using the point slope form

$\begin{array}{l}y−y_1=m\left(x−x_1\right)\\ y−2=\frac{−1}{3}\left(x−\left(2\right)\right)\\ y−2=\frac{−1}{3}(x−2)\\ y−2=\frac{−1}{3}x+\left(−\frac{1}{3}\right)\left(−2\right)\end{array}$

Simplifying further,

$\begin{array}{c}y−2=−\frac{1}{3}x+\frac{2}{3}\\ \text{Add2onbothsides}\\ y−2+2=\frac{−1}{3}x+\frac{2}{3}+2\\ y=\frac{−1}{3}x+\frac{2+2\left(3\right)}{3}(\text{LCMof1and3is3)}\end{array}$

Again simplifying further,

$\begin{array}{l}y=\frac{−1}{3}x+\frac{2+6}{3}\\ y=\frac{−1}{3}x+\frac{8}{3}\end{array}$

This is in slope intercept form.