Q. 29. - Algebra 2

Question

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

passes through $\left(-\mathbf{2},\mathbf{5}\right)$ and $\left(\mathbf{3},\mathbf{1}\right)$

Step-by-Step Solution

Answer

The equation of the required straight line in slope-intercept form is $y=-\frac{4}{5}x+\frac{17}{5}$.

Step 1. State the concept

The slope intercept form of a straight-line equation is $y=mx+c$ where m is the slope and c is the y-intercept.

The equation of a straight-line passing through the points $\left(h,k\right)$ and $\left(a,b\right)$ is given as $\frac{y-k}{x-h}=\frac{b-k}{a-h}$.

Step 2. List the given data

It is given that the line passes through $\left(-2,5\right)$ and $\left(3,1\right)$.

Then, $\left(h,k\right)=\left(-2,5\right)$ and $\left(a,b\right)=\left(3,1\right)$.

Step 3. Write the equation

Put $\left(h,k\right)=\left(-2,5\right)$ and $\left(a,b\right)=\left(3,1\right)$ in $\frac{y-k}{x-h}=\frac{b-k}{a-h}$ to get,

 

$\frac{y-5}{x-\left(-2\right)}=\frac{1-5}{3-\left(-2\right)}$

 

$\frac{y-5}{x+2}=\frac{-4}{3+2}$  (Simplify)

 

$\frac{y-5}{x+2}=-\frac{4}{5}$  (Simplify)

 

$\frac{y-5}{x+2}\left(x+2\right)=-\frac{4}{5}\left(x+2\right)$   (Multiply both sides by $x+2$)

 

$y-5=-\frac{4}{5}\left(x+2\right)$  (Simplify)

 

$y-5=-\frac{4}{5}x-\frac{8}{5}$  (Distributive property)

 

$y-5+5=-\frac{4}{5}x-\frac{8}{5}+5$  (Add 5 to both sides)

 

$y=-\frac{4}{5}x+\frac{17}{5}$  (Simplify)

 

So, the required equation of the straight line in slope-intercept form is $y=-\frac{4}{5}x+\frac{17}{5}$.