Q28. - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions

$28. \text{ passes through }\left(-6,15\right),\text{ parallel to the graph of }2x+3y=1$

Step-by-Step Solution

Answer

The slope intercept form of the line which passes through the point $\left(-6,15\right)$ and parallel to the graph of $2x+3y=1$ is $y=-\frac{2}{3}x+11$. 

Step-1 – Apply the concept of slope-intercept form and point slope form

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 m is the slope and b is 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.

Step-2 – Convert the equation of the given line into slope-intercept form

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,

$\begin{array}{c}2x+3y=1\\ \Rightarrow 2x+3y-2x=1-2x(\text{subtract }2x\text{ from both sides)}\\ \Rightarrow 3y=1-2x\end{array}$

Divide both sides by 3,

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

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

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

Step-3 – Find the equation using point-slope form

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,

$\begin{array}{c}y-15=-\frac{2}{3}x-\frac{12}{3}\\ y-15=\frac{-2}{3}x-4\\ \text{Add 15 on both sides}\\ y-15+15=\frac{-2}{3}x-4+15\\ y=\frac{-2}{3}x+11\end{array}$

This is in slope intercept form.