Question

What is the solution to this system of equations?

x + 3y - z = 6
4x - 2y + 2z= -10
6x + z= -12

(an explanation would be great because i barely understand how to do this)

Answer 1

`x=-3, y=5, z=6`

Explanation:

1. Divide by 2 both sides in the second equation:

• 
  `4x - 2y + 2z= -10`
• 
  `2x - y + z= -5`

2. From the third equation find z in terms of x:

• 
  `6x + z= -12`
• 
  `z= -6x-12`

3. Add up the first and new second equations and find y in terms of x:

• 
  `(x+2x)+(3y-y)+(-z+z)=6+(-5)`
• 
  `3x+2y=1`
• 
  `2y=1-3x`
• 
  `y=0.5-1.5x`

4. Put new alternatives instead of y and z in the first equation:

• 
  `x+3(0.5-1.5x)-(-6x-12)=6`
• 
  `x+1.5-4.5x+6x+12=6`
• 
  `2.5x=-7.5`
• 
  `x=-3`

5. Now we can easily find other two unknowns by using their equations by means of x:

• 
  `y=0.5-1.5*(-3)`
• 
  `y=0.5+4.5`
• 
  `y=5`

• 
  `z=-6*(-3)-12`
• 
  `z=18-12`
• 
  `z=6`

Answer 2

Explanation:

2x + 6y - 2z = 12

12x - 6y + 6z = -30

14x + 4z = -18

6x + z = -12

14x + 4z = -18

-24x - 4z = 48

-10x = 30

x = -3

6(-3) + z = -12

-18 + z = -12

z = 6

-3 + 3y - 6 = 6

-9 + 3y = 6

3y = 15

y = 5

x = -3, y = 5, z = 6