+752 The three points (3,-5), (-a + 2, 3), and (2a+3,2) lie on the same line. What is a?
+9466 Since all the points are on the same line, the slope between each point will be the same.
slope = \(\frac{\text{change in y}}{\text{change in x}}\)
slope between first and second points = \(\frac{(-5)-(3)}{(3)-(-a+2)}=\frac{-8}{1+a} \)
slope between second and third points = \(\frac{(3)-(2)}{(-a+2)-(2a+3)}=\frac{1}{-3a-1} \)
slope between third and first points = \(\frac{(2)-(-5)}{(2a+3)-(3)}=\frac{7}{2a}\)
Let's pick any two and equate them.
\(\frac{7}{2a}=\frac{-8}{1+a} \) Cross - multiply...
(7)(1+a) = (-8)(2a)
7 + 7a = -16a
7 = -23a
-7/23 = a And here is a graph: https://www.desmos.com/calculator/2pdpfrqz05
+26367 The three points (3,-5), (-a + 2, 3), and (2a+3,2) lie on the same line.
What is a?
Intercept theorem:
\(\begin{array}{|rcll|} \hline \dfrac{(2a-3)-(3)}{ 2-(-5) } &=& \dfrac{ (-a+2) - 3 } { 3-(-5) } \\\\ \dfrac{2a}{ 7 } &=& \dfrac{ -a -1 } { 8 } \\\\ 16a &=& -7a-7 \\ 23a &=& -7 \\ \mathbf{a} &\mathbf{=}& \mathbf{ -\frac{7}{23} } \\ \hline \end{array} \)
