Answer:
Point-slope form:
[tex]y-4=\displaystyle\frac{1}{3}(x+4)[/tex]
[tex]y-8=m(x-8)[/tex]
Slope-intercept form:
[tex]y=\displaystyle\frac{1}{3}x+\displaystyle\frac{16}{3}[/tex]
Standard form:
[tex]-x+3y=16[/tex]
Step-by-step explanation:
Hi there!
1) Point-slope form
Point-slope form: [tex]y-y_1=m(x-x_1)[/tex] where [tex](x_1,y_1)[/tex] is a point on the line and m is the slope
First, determine the slope:
[tex]m=\displaystyle\frac{y_2-y_1}{x_2-x_1}[/tex] where two points on the line are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
Plug in the points (-4,4) and (8,8):
[tex]m=\displaystyle\frac{8-4}{8-(-4)}\\\\m=\displaystyle\frac{8-4}{8+4}\\\\m=\displaystyle\frac{4}{12}\\\\m=\displaystyle\frac{1}{3}[/tex]
Therefore, the slope of the line is [tex]\displaystyle\frac{1}{3}[/tex]. Plug this into [tex]y-y_1=m(x-x_1)[/tex]:
[tex]y-y_1=\displaystyle\frac{1}{3}(x-x_1)[/tex]
Now, for [tex](x_1,y_1)[/tex], we can either plug in (-4,4) or (8,8):
[tex]y-4=\displaystyle\frac{1}{3}(x-(-4))\\\\y-4=\displaystyle\frac{1}{3}(x+4)[/tex]
or
[tex]y-8=m(x-8)[/tex]
2) Slope-intercept form
Slope-intercept form: [tex]y=mx+b[/tex] where m is the slope and b is the y-intercept (the value of y when the line crosses the y-axis)
[tex]y=mx+b[/tex]
Plug in the slope
[tex]y=\displaystyle\frac{1}{3}x+b[/tex]
Now, to determine the y-intercept, plug in one of the points (-4,4) or (8,8) and solve for b:
[tex]8=\displaystyle\frac{1}{3}(8)+b\\\\8=\displaystyle\frac{8}{3}+b\\\\8-\displaystyle\frac{8}{3}=b\\\\b=\frac{16}{3}[/tex]
Therefore, the y-intercept is [tex]\displaystyle\frac{16}{3}[/tex]. Plug this back into [tex]y=\displaystyle\frac{1}{3}x+b[/tex]:
[tex]y=\displaystyle\frac{1}{3}x+\displaystyle\frac{16}{3}[/tex]
3) Standard form
Standard form:
[tex]Ax+By=C[/tex] where A, B, and C are numbers which are typically integers
[tex]y=\displaystyle\frac{1}{3}x+\displaystyle\frac{16}{3}[/tex]
Organize this into standard form:
[tex]-\displaystyle\frac{1}{3}x+y=\displaystyle\frac{16}{3}[/tex]
Multiply both sides by 3 so A, B, and C are integers:
[tex]-x+3y=16[/tex]
I hope this helps!
