A line passes through the points $A,$ $B,$ and $C.$ If $A = (-2,5),$ $B = (-3,1),$ and $C = (4,y),$ then find $y.$

Answer 1

We proceed as follows. Since a line passes through $(x_1, y_1)$ and $(x_2, y_2),$ we know that the slope of the line is
$$\frac{y_2-y_1}{x_2-x_1}.$$Hence, the slope of the line passing through $A$ and $B$ is
$$\frac{1-5}{-3-(-2)}=1,$$so the line is of the form $y - 5 = 1 (x + 2).$ Also, $C = (4,y)$ must lie on this line, so substituting in, we get
$$y-5=x+2.$$Therefore, $y = x + 7.$

Thus, the $y$-coordinate of $C$ is $\boxed{11}.$