Question

Find X and Y, if `2X + 3Y = [(2,3),(4,0)] and 3X + 2Y = [(2, -2),(-1,5)]`

Answer 1

`2"X" + 3"Y" = [(2,3),(4,0)]` ...(i)

`3"X" + 2"Y" = [(2, -2),(1-,5)]` ...(ii)

Subtracting equation (i) by 3 and (ii) by 2,

3 (2x + 3y) - 2 (3x + 2y)

`= 3 [(2,3),(4,0)] - 2 [(2,-2),(-1,5)]`

`=> 6 "x" + 9 "y" - 6"x" - 4 "y"`

`= [(6,9),(12, 0)] - [(4, -4),(-2,10)]`

`=> 5 "y" = [(2,13),(14,-10)]`

`=> "y" = [(2/5,13/5),(14/5,-2)]`

Again, subtracting equation (i) multiplied by 2 and (ii) by 3,

2 (2x + 3y) - 3 (3x - 2y)

`= 2 [(2,3),(4,0)] - 3 [(2,-2),(-2,5)]`

`4"x" - 9"y" = [(4,6),(8,0)] - [(6,-6),(-3,15)]`

`-5"x" = [(-2,12),(11,-15)]`

`"x" = -1/5 [(-2,12),(11,-15)]`

`= [(2/5,-12/5),(-11/5,3)]`

Hence, `x= [(2/5,-12/5),(-11/5,3)], "y" = [(2/5,13/5),(14/5,-2)]`