Question

find an equation for the line that passes through the points (-4, -1) and (2, 3).

Explanation:

Step1: Calculate the slope

The slope $m$ formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(-4,-1)$ and $(x_2,y_2)=(2,3)$. Then $m=\frac{3-(-1)}{2-(-4)}=\frac{3 + 1}{2 + 4}=\frac{4}{6}=\frac{2}{3}$.

Step2: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using the point $(2,3)$ and $m = \frac{2}{3}$, we have $y - 3=\frac{2}{3}(x - 2)$.

Step3: Convert to slope - intercept form

Expand the right side: $y-3=\frac{2}{3}x-\frac{4}{3}$. Then add 3 to both sides: $y=\frac{2}{3}x-\frac{4}{3}+3=\frac{2}{3}x+\frac{- 4 + 9}{3}=\frac{2}{3}x+\frac{5}{3}$.

Step4: Convert to general form

Multiply through by 3 to get $3y = 2x+5$, and then rearrange to $2x-3y+5 = 0$.

Answer:

$y=\frac{2}{3}x+\frac{5}{3}$ (slope - intercept form) or $2x - 3y+5=0$ (general form)