### Instruction: The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line.  What is $a$?### Response: To check if three points are collinear, I can use the slope formula to see if the slopes between any two pairs of points are equal.
Let me call the points $A$, $B$, and $C$ for convenience.
Then the slope of $AB$ is $\frac{3-(-5)}{-a+2-3} = \frac{8}{-a-1}$, and the slope of $AC$ is $\frac{2-(-5)}{2a+3-3} = \frac{7}{2a}$.
So I need to solve the equation $\frac{8}{-a-1} = \frac{7}{2a}$ for $a$.
I can cross-multiply to get $16a = -7a - 7$, and then simplify to get $23a = -7$.