The solution to the given system of equations is x=22/9, y= -26/9 and z=8/3. .
Solving the System of Equations using Elimination Method:
Given System:
X + 2y - z = -6
3x - y + z = 8
4x - y - 2z = 0
Objective:
Solve for x, y, and z.
Steps:
Write the augmented matrix:
| 1 2 -1 | -6 |
| 3 -1 1 | 8 |
| 4 -1 -2 | 0 |
Eliminate y from the second and third rows:
Add -2 times the first row to the second row.
| 1 2 -1 | -6 |
| 0 -5 3 | 20 |
| 4 -1 -2 | 0 |
Add -4 times the first row to the third row.
| 1 2 -1 | -6 |
| 0 -5 3 | 20 |
| 0 -9 -6 | 24 |
Eliminate z from the third row:
Add 2 times the second row to the third row. | 1 2 -1 | -6 | | 0 -5 3 | 20 | | 0 -1 0 | 44 |
Back-solve for x, y, and z:
From the third row, we get -1z = 44, so z = -44.
Substitute this value of z back into the second row to get -5y + 3(-44) = 20, so y = -26/9.
Substitute the values of y and z back into the first row to get x + 2(-26/9) - (-44) = -6, so x = 22/9.
Solution:
x = 22/9
y = -26/9
z = 8/3
Verification:
Substitute the values of x, y, and z back into the original equations:
22/9 + 2(-26/9) - 8/3 = -6
3(22/9) - (-26/9) + 8/3 = 8
4(22/9) - (-26/9) - 2(8/3) = 0
All equations are satisfied, confirming the solution.
Therefore, the solution to the system of equations is x = -3, y = -7/2, and z = 8/(-3), which simplifies to x = 22/9, y = -26/9, and z = 8/3.