solve the system 3x16x29x33x50 4x18x212x34x50 3x16x29x34x45x

solve the system

3x1+6x2+9x3-3x5=0
4x1+8x2+12x3-4x5=0
-3x1-6x2-9x3+4x4-5x5=12
-2x1-4x2-6x3+x4=3

3x1+6x2+9x3-3x5=0            (1)
4x1+8x2+12x3-4x5=0          (2)
-3x1-6x2-9x3+4x4-5x5=12    (3)
-2x1-4x2-6x3+x4=3             (4)

eq(1) +eq(2) we get

3x1+6x2+9x3-3x5=0

-3x1-6x2-9x3+4x4-5x5=12

=> 4x4 -8x5 = 12

4x4 -8x5 +8x5 = 12 +8x5

4x4 = 12 +8x5

x4 = 3 +2x5 (5)

eq(2) + 2*eq(4)

4x1+8x2+12x3-4x5=0

-4x1-8x2-12x3+2x4=6

we get 2x4 -4x5 =6

or x4 = 3 + 2x5

we can see that all other variables(x1,x2,x3) are free

from (1)

3x1 +6x2 +9x3 -3x5 =0

3x1 +6x2 +9x3 = 3x5 // we have added 3x5 on both sides

x5 = x1 +2x2 +3x3 // divided both sides by 3

from eq(5)

x4 = 3 +2x5

or x4 = 3 + 2(x1 +2x2 +3x3)

x4 = 3 +2x1 +4x2 +6x3 // we can assume x1 =a , x2 =b and x3 =c