Question

Three vertices of parallelogram PRTV are P(-4,-4), R(-10,0), and V(5,-1). Find the coordinates of vertex T.

Answer 1

Slope of TV = Slope of PR
(y + 1)/(x - 5) = (0 + 4)/(-10 + 4)
(y + 1)/(x - 5) = -4/6
6(y + 1) = -4(x - 5)
6y + 6= -4x + 20
equation #1:  6y + 4x = 14

Slope of RT = Slope of PV
(y - 0)/(x + 10) = (-1 + 4)/(5 + 4)
(y - 0)/(x + 10) = 3/9
3(y - 0) = 9(x + 10)
3y  = 9x +90
equation #2:  3y  = 9x +90

We now we have 2 equations with 2 variables, and we can use any method (substitution, linear combination, matrix) to solve for the x and y coordinates. 

equation #1:  6y + 4x = 14
equation #2:  3y - 9x  = 90

Multiply equation 2 by -2

equation #1:    6y + 4x = 14
equation #2:  -6y + 18x  = -180

Add together Eq. 1 & 2

22x = -166
x = -7.5

6y +4(-7.5) = 14
6y = 44
y = 7 1/3

(-7 1/2, 7 1/3)