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 $2x+3y=1$.

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,

$\begin{array}{c}2x+3y=1\\ ⇒2x+3y−2x=1−2x(\text{subtract}2x\text{frombothsides)}\\ ⇒3y=1−2x\end{array}$

Divide both sides by 3,

$\begin{array}{c}⇒\frac{3y}{3}=\frac{1−2x}{3}\\ ⇒y=\frac{1}{3}−\frac{2x}{3}\\ ⇒y=−\frac{2}{3}x+\frac{1}{3}\end{array}$

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

Hence, the slope is $m=\frac{-2}{3}$

Given the point $\left(-6,15\right)$ and slope is $m=-\frac{2}{3}$.

Therefore by using the point slope form

$\begin{array}{l}y−y_1=m\left(x−x_1\right)\\ y−15=\frac{−2}{3}\left(x−\left(−6\right)\right)\\ y−15=\frac{−2}{3}(x+6)\\ y−15=\frac{−2}{3}x+\left(−\frac{2}{3}\right)\left(6\right)\end{array}$

Simplifying further,

$\begin{array}{c}y−15=−\frac{2}{3}x−\frac{12}{3}\\ y−15=\frac{−2}{3}x−4\\ \text{Add15onbothsides}\\ y−15+15=\frac{−2}{3}x−4+15\\ y=\frac{−2}{3}x+11\end{array}$

This is in slope intercept form.