Answer:
(a)
[tex]b = 6[/tex] and [tex]a = 5[/tex]
(b) [tex]x = 1[/tex]
Step-by-step explanation:
Given
[tex](x_1,y_1) = (2,2)[/tex]
[tex](x_2,y_2) = (4,b)[/tex]
[tex](x_3,y_3) = (a,8)[/tex]
[tex](x_4,y_4) = (6,10)[/tex]
See attachment
Solving (a): The values of a and b
First, calculate the slope of the line:
[tex]m = \frac{y_c - y_d}{x_c - x_d}[/tex]
Let c = 4 and d = 1
[tex]m = \frac{y_4 - y_1}{x_4 - x_1}[/tex]
So, we have:
[tex]m = \frac{10 - 2}{6 - 2}[/tex]
[tex]m = \frac{8}{4}[/tex]
[tex]m = 2[/tex]
To solve for the value of b, we apply slope formula
[tex]\frac{y_c - y_d}{x_c - x_d} = m[/tex]
Let c = 2 and d = 1
So, we have:
[tex]\frac{y_2 - y_1}{x_2 - x_1} = m[/tex]
Substitute 2 for m and [tex](x_1,y_1) = (2,2)[/tex] ; [tex](x_2,y_2) = (4,b)[/tex]
[tex]\frac{b - 2}{2} = 2[/tex]
Multiply both sides by 2
[tex]2 * \frac{b - 2}{2} = 2 * 2[/tex]
[tex]b - 2 = 4[/tex]
Add 2 to both sides
[tex]b - 2 +2= 4 + 2[/tex]
[tex]b = 6[/tex]
To solve for the value of a, we apply slope formula
[tex]\frac{y_c - y_d}{x_c - x_d} = m[/tex]
Let c = 3 and d = 1
[tex]\frac{y_3 - y_1}{x_3 - x_1} = m[/tex]
Substitute 2 for m and [tex](x_1,y_1) = (2,2)[/tex] ; [tex](x_3,y_3) = (a,8)[/tex]
[tex]\frac{8 - 2}{a- 2} = 2[/tex]
[tex]\frac{6}{a- 2} = 2[/tex]
Cross Multiply
[tex]2(a-2) = 6[/tex]
Open bracket
[tex]2a-4 = 6[/tex]
Add 4 to both sides
[tex]2a-4+4 = 6+4[/tex]
[tex]2a= 10[/tex]
Divide both sides by 2
[tex]a = 5[/tex]
Solving (b): The value of y when x = 0.
This point is represented as: [tex](x_5,y_5) = (x,0)[/tex]
Apply slope formula
[tex]\frac{y_5 - y_1}{x_5 - x_1} = m[/tex]
Substitute 2 for m and [tex](x_1,y_1) = (2,2)[/tex] [tex](x_5,y_5) = (x,0)[/tex]
[tex]\frac{0 - 2}{x - 2} = 2[/tex]
[tex]\frac{- 2}{x - 2} = 2[/tex]
[tex]-\frac{2}{x - 2} = 2[/tex]
Cross Multiply
[tex]-2 = 2 * (x - 2)[/tex]
[tex]-2 = 2 x - 4[/tex]
Collect Like Terms
[tex]4 - 2 = 2x[/tex]
[tex]2 = 2x[/tex]
Divide both sides by 2
[tex]1 = x[/tex]
[tex]x = 1[/tex]
