solve by Gauss elimination
x+y-z=2
-x+3y +z=3
4x-y+3z= 5
show work for each step please
Please show some work so we know where your problem is.
If you have no clue at all, review your class notes or try the following link:
http://en.wikipedia.org/wiki/Gaussian_elimination
To solve the given system of equations using Gaussian elimination, we'll perform a series of row operations to transform the system into row-echelon form. Here are the step-by-step calculations:
Step 1: Write the given system of equations in matrix form:
| 1 1 -1 | | x | | 2 |
|-1 3 1 | x | y | = | 3 |
| 4 -1 3 | | z | | 5 |
Step 2: Row 2 → Row 2 + Row 1 (to eliminate x in Row 2)
| 1 1 -1 | | x | | 2 |
| 0 4 0 | x | y | = | 5 |
| 4 -1 3 | | z | | 5 |
Step 3: Row 3 → Row 3 - 4 * Row 1 (to eliminate x in Row 3)
| 1 1 -1 | | x | | 2 |
| 0 4 0 | x | y | = | 5 |
| 0 -5 7 | | z | | -3 |
Step 4: Row 3 → Row 3 + (5/4) * Row 2 (to eliminate y in Row 3)
| 1 1 -1 | | x | | 2 |
| 0 4 0 | x | y | = | 5 |
| 0 0 7 | | z | | 14/4 |
Step 5: Row 3 → (1/7) * Row 3 (to make the leading coefficient of z 1)
| 1 1 -1 | | x | | 2 |
| 0 4 0 | x | y | = | 5 |
| 0 0 1 | | z | | 2 |
Step 6: Row 1 → Row 1 - Row 3 (to eliminate z in Row 1)
| 1 1 0 | | x | | 0 |
| 0 4 0 | x | y | = | 5 |
| 0 0 1 | | z | | 2 |
Step 7: Row 2 → (1/4) * Row 2 (to make the leading coefficient of y 1)
| 1 1 0 | | x | | 0 |
| 0 1 0 | x | y | = | 5/4 |
| 0 0 1 | | z | | 2 |
Step 8: Row 1 → Row 1 - Row 2 (to eliminate y in Row 1)
| 1 0 0 | | x | | -5/4 |
| 0 1 0 | x | y | = | 5/4 |
| 0 0 1 | | z | | 2 |
Solution: x = -5/4, y = 5/4, z = 2