Q29. - Algebra 2

Question

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

$\text{ passes through }\left(2,2\right),\text{ parallel to the graph of }x+3y=7$

Step-by-Step Solution

Answer

The slope intercept form of the line which passes through the point $\left(2,2\right)$ and parallel to the graph of $x+3y=7$ is $y=-\frac{1}{3}x+\frac{8}{3}$.

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 $x+3y=7$.

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,

$\begin{array}{c}x+3y=7\\ \Rightarrow x+3y-x=1-x(\text{subtract }x\text{ from both sides)}\\ \Rightarrow 3y=7-x\end{array}$

Divide both sides by 3,

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

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

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

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

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,

$\begin{array}{c}y-2=-\frac{1}{3}x+\frac{2}{3}\\ \text{Add 2 on both sides}\\ y-2+2=\frac{-1}{3}x+\frac{2}{3}+2\\ y=\frac{-1}{3}x+\frac{2+2\left(3\right)}{3}(\text{LCM of 1 and 3 is 3)}\end{array}$

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.