what is the answer?

Question

If the line passing through the points (2, a) and (5, - 3) is parallel to the line passing through the points (4, 8) and (- 5, a + 1) , what is the value of a?

Answer 1

If two lines are parallel, then the slopes of the lines are equal.

 

The slope of the line is the change in the y coordinate over the change in the x coordinate

 

For the first line, this is ( -3 -a)/(5 -2).  For the 2nd line, this is (a + 1 - 8)/(-5 -4);  simplifying and setting equal to each other, we get:

 

(-3 - a)/3 = (a-7)/-9

 

let's cross multiply and we get

 

-9 * (-3 - a) = 3 * (a - 7)

 

divide each side by 3

 

-3 (-3 - a) = a - 7

 

simplify:  9 + 3a = a - 7

 

add 7  to each side and subtract a from each side

 

16 + 2a  = 0

 

subtract 2a from each side

 

16 = - 2*a

 

divide each side by -2

 

-8 = a

 

If you go back to the initial equation and plug in -8 for a, you show that each side = 5/3 

Answer 2

If the lines are parallel, then the slopes have to be the same.

 

So, (a + 1 - 8)/ (-5 - 4) = (-3 - a)/(5 - 2)

 

      (a - 7)/(-9) = (-3 - a)/3

 

Cross multiply to get:  3(a - 7) = -9(-3 - a)

                                  3a - 21 = 27 + 9a

                                         -6a = 48

                                            a = -8