Simultaneous Linear Questions in Two Variables. (Difficult)

1. 2(x+y)=y+10

    4(x+y)=5y+32 

Simplify

2x + 2y  = y + 10   ⇒   2x  +  y  =  10  ⇒ y = 10 - 2x    (1)

4x + 4y = 5y + 32 ⇒    4x - y  = 32    (2)

Sub (1)  into (2)  for y

4x - (10 - 2x) = 32

6x - 10   = 32

6x = 42

x  = 7

And using (1)    y = 10 - 2(7)  =  -4

So   (x, y)  =  (7, -4)

2.

7(4-x)=2(y-3)    ⇒  28 - 7x = 2y - 6 ⇒  2y + 7x  =  34    (1)

4(x-y)=7-y ⇒  4x - 4y = 7 - y  ⇒  3y - 4x = - 7      (2)

Multiply (1) by 3   and (2) by -2

6y + 21x  = 102

-6y + 8x   =   14             add these

29x = 116         divide both sides by 29

x = 4

And using  (1) to find y

2y + 7(4) = 34

2y + 28 = 34

2y = 6

y = 3

So   (x, y)  =  (4, 3)