Let,
x₁, y₁ = 2, 2
x₂, y₂ = 6, 10
a.) The slope of the line.
[tex]\text{ Slope = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\text{ = }\frac{\text{ 10 - 2}}{\text{ 6 - 2}}\text{ = }\frac{\text{ 8}}{\text{ 4}}[/tex][tex]\text{ Slope = 2}[/tex]Therefore, the slope of the line is 2.
b.) The y-intercept of the line.
Substitute slope = m = 2 and x, y = 2, 2 in y = mx + b
[tex]\text{ y = mx + b}[/tex][tex]\text{ 2 = 2(2) + b}[/tex][tex]\text{ 2 = 4 + b }\rightarrow\text{ b = 2 - 4}[/tex][tex]\text{ b = y-intercept = -2}[/tex]Therefore, the y-intercept is -2.
For us to answer the other 2 questions, let's first complete the equation of the graph.
Substitute slope = 2 and y-intercept = -2 in the y = mx + b
y = mx + b
y = (2)x + (-2)
y = 2x - 2
The equation of the line is y = 2x - 2
c.) Finding the value of a.
x = a
y = 8
We get,
[tex]\text{ y = 2x - 2}[/tex][tex]\text{8 = 2a - 2}[/tex][tex]\text{ 2a = 8 + 2 = 10}[/tex][tex]\text{ }\frac{\text{2a}}{\text{ 2}}\text{ = }\frac{\text{10}}{\text{ 2}}[/tex][tex]\text{ a = 5}[/tex]Therefore a = 5
d.) Finding the value of b.
x = 4
y = b
[tex]\text{ y = 2x - 2}[/tex][tex]\text{ b = 2(4) - 2}[/tex][tex]\text{ b = 8 - 2 = 6}[/tex]Therefore, b = 6