Answer:
[tex]x=\frac{83}{50}[/tex]
Step-by-step explanation:
we know that
If the three points are collinear
then
[tex]m_A_B=m_A_C[/tex]
we have
A (1, 2/3), B (x, -4/5), and C (-1/2, 4)
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
step 1
Find the slope AB
we have
[tex]A(1,\frac{2}{3}),B(x,-\frac{4}{5})[/tex]
substitute in the formula
[tex]m_A_B=\frac{-\frac{4}{5}-\frac{2}{3}}{x-1}[/tex]
[tex]m_A_B=\frac{\frac{-12-10}{15}}{x-1}[/tex]
[tex]m_A_B=-\frac{22}{15(x-1)}[/tex]
step 2
Find the slope AC
we have
[tex]A(1,\frac{2}{3}),C(-\frac{1}{2},4)[/tex]
substitute in the formula
[tex]m_A_C=\frac{4-\frac{2}{3}}{-\frac{1}{2}-1}[/tex]
[tex]m_A_C=\frac{\frac{10}{3}}{-\frac{3}{2}}[/tex]
[tex]m_A_C=-\frac{20}{9}[/tex]
step 3
Equate the slopes
[tex]m_A_B=m_A_C[/tex]
[tex]-\frac{22}{15(x-1)}=-\frac{20}{9}[/tex]
solve for x
[tex]15(x-1)20=22(9)[/tex]
[tex]300x-300=198[/tex]
[tex]300x=198+300[/tex]
[tex]300x=498[/tex]
[tex]x=\frac{498}{300}[/tex]
simplify
[tex]x=\frac{83}{50}[/tex]
