Linear equations are typically organized in slope-intercept form:
[tex]y=mx+b[/tex]
To determine a linear equation in slope-intercept form:
We're given:
First, find the slope of the line (m):
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex] where two points that fall on the line are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
⇒ Plug in the given points (-1,-4) and (5,-2):
[tex]m=\dfrac{-2-(-4)}{5-(-1)}\\\\m=\dfrac{-2+4}{5+1}\\\\m=\dfrac{2}{6}\\\\m=\dfrac{1}{3}[/tex]
⇒ Therefore, the slope of the line is [tex]\dfrac{1}{3}[/tex]. Plug this into y=mx+b:
[tex]y=\dfrac{1}{3}x+b[/tex]
Now, determine the y-intercept (b):
[tex]y=\dfrac{1}{3}x+b[/tex]
⇒ Plug in one of the points and solve for b:
[tex]-4=\dfrac{1}{3}(-1)+b\\\\-4=-\dfrac{1}{3}+b\\\\-4+\dfrac{1}{3}=b\\\\b=-\dfrac{11}{3}[/tex]
⇒ Therefore, the y-intercept is [tex]-\dfrac{11}{3}[/tex]. Plug this back into our original equation:
[tex]y=\dfrac{1}{3}x-\dfrac{11}{3}[/tex]
[tex]y=\dfrac{1}{3}x-\dfrac{11}{3}[/tex]
