Find $a$ if the point $(3,a)$ is on the line that passes through $(-2,7)$ and $(5,-3)$.

the slope between  (-2, 7)  and  (5, -3)   =   [ -3 - 7 ] / [ 5 - -2 ]   =   -10/7

 

And since  (3, a)  lies on the same line as those two points, then we know.....

 

the slope between  (3, a)  and  (-2, 7)   =   -10/7

 

[ 7 - a ] / [ -2 - 3 ]   =   -10/7

 

[ 7 - a ] / [ -5 ]   =   -10/7            Multiply both sides by  -5 .

 

7 - a   =   50/7                           Subtract  7  from both sides.

 

-a   =   50/7 - 7

 

-a   =   50/7 - 49/7

 

-a   =   1/7

 

 a   =   -1/7                    And here's a graph:  https://www.desmos.com/calculator/prq2oggmcv