Question

Please solve the following simultaneous equation: 4x-3y=15 8x^2-27y^2=45

Answer 1

{ 4 x - 3y = 15
  8 x² - 27 y² =45

Isolate y for 4 x - 3y = 15

4 x - 3y = 15 ( add 3y to both sides ):

4x-3y +3y = 15 + 3y

4x =  15 + 3y

Divide both sides by 4 :

4x / 4 = 15 + 3y / 4

x = 15+3y / 4

Plug  the solutions x = 15+3y / 4 into : 8 x² - 27 y²=45

8 ( 15+3y / 4 )² - 27 y² = 45

1/2 ( 3y+15)² - 27 y² = 45

Multiply  by LCD = 2

9 ( y+5)² - 54 y² = 90

-45 y² +  90 y  + 225 = 90

- 45 y² + 90 y = 225 - 90

-45 y² + 90 y = 135 = 0

-45 ( y + 1 ) ( y - 3 ) = 0

Using the zero factor principle:

Solve ( y + 1 ) = 0

y = - 1

Solve ( y - 3 ) = 0

y = 3

y = -1 and y = 3

Plug the solution y = -1 , y = 3 into : 4 x - 3 y = 15

4 x - 3 y =  15  for - 1 : 

4 x - 3 ( - 1 ) = 15

4 x + 3 = 15

4 x = 15 - 3

4 x = 12

x = 12 / 4

x = 3

For 3 :

4 x - 3 y  = 15

4 x - 3 ( 3 ) = 15

4 x - 9 = 15

4 x = 15 + 9

4 x = 24

x = 24 / 4

x = 6

Therefore the final solution :

y = 3 ,  x = 6
y = - 1 , x  = 3

hope this helps!