Answer:
[tex]\textsf{Line \#2}: \quad y=x-5[/tex]
[tex]\textsf{Line \#3}: \quad y=\dfrac{1}{4}(x-1)-4[/tex]
[tex]\textsf{Line \#4}: \quad y=-8x-29[/tex]
Step-by-step explanation:
Given points:
- (-4, 3)
- (6, 1)
- (1, -4)
- (-3, -5)
To find the equations for each of the lines:
- Find the slope of the line by substituting two points on the line into the slope formula.
- Substitute the found slope and one of the points on the line into the point-slope formula and simplify.
Line #2
Points:
- Let (x₁, y₁) = (6, 1)
- Let (x₂, y₂) = (1, -4)
Find the slope:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-4-1}{1-6}=\dfrac{-5}{-5}=1[/tex]
Substitute the found slope and one of the points into the point-slope formula:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-1=1(x-6)[/tex]
[tex]\implies y=(x-6)+1[/tex]
[tex]\implies y=x-5[/tex]
Line #3
Points:
- Let (x₁, y₁) = (1, -4)
- Let (x₂, y₂) = (-3, -5)
Find the slope:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-5-(-4)}{-3-1}=\dfrac{-1}{-4}=\dfrac{1}{4}[/tex]
Substitute the found slope and one of the points into the point-slope formula:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-(-4)=\dfrac{1}{4}(x-1)[/tex]
[tex]\implies y+4=\dfrac{1}{4}(x-1)[/tex]
[tex]\implies y=\dfrac{1}{4}(x-1)-4[/tex]
Line #4
Points:
- Let (x₁, y₁) = (-3, -5)
- Let (x₂, y₂) = (-4, 3)
Find the slope:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{3-(-5)}{-4-(-3)}=\dfrac{8}{-1}=-8[/tex]
Substitute the found slope and one of the points into the point-slope formula:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-(-5)=-8(x-(-3))[/tex]
[tex]\implies y+5=-8(x+3)[/tex]
[tex]\implies y=-8(x+3)-5[/tex]
[tex]\implies y=-8x-24-5[/tex]
[tex]\implies y=-8x-29[/tex]
