The given equations are
- 5x + 5y - 5z = 0
10x + 2y - 5z = - 6
- 5x + 2y + 3z = 10
Let us divide through the first equation by 5 and make y the subject of the equation. we have
- 5x/5 + 5y/5 - 5z/5 = 0/5
- x + y - z = 0
We would add x and z to both sides of the equation. We have
- x + x + y - z + z = 0 + x + z
y = x + z
The next step is to substitute y = x + z into the second and third equations. Substituting y = x + z into the second equation, we have
10x + 2(x + z) - 5z = - 6
By expanding the parentheses on the left, we have
10x + 2x + 2z - 5z = - 6
8x - 3z = - 6 equation 4
Substituting y = x + z into the third equation, we have
- 5x + 2(x + z) + 3z = 10
By expanding the parentheses on the left, we have
- 5x + 2x + 2z + 3z = 10
- 3x + 5z = 10 equation 5
We would solve equations 4 and 5 by using the method of elimination. We would eliminate x by multiplying equation 4 by 3 and equation 5 by 8. The new equations would be
24x - 9z = - 18
- 24x + 40z = 80
We would add both equations. We have
24x - 24x - 9z + 40z = - 18 + 80
31z = 62
Dividing both sides of the equation by 31,
31z/31 = 62/31
z = 2
Substituting z = 2 into 8x - 3z = - 6, we have
8x - 3(2) = - 6
8x - 6 = - 6
Sdding 6 to both sides of the equation, we have
8x - 6 + 6 = - 6 + 6
8x = 0
Dividing both sides of the equation by 8,
8x/8 = 0/8
x = 0
Finally, we would substitute x = 0 and z = 2 into y = x + z. We have
y = 0 + 2
y = 2
The solution is
(0, 2, 2)