Find the solution of the following system of equations using Gaussian elimination method and show the steps. x-y+z = 4 5x-y+z = 6 3x-y+5z= 5

Sure, let's solve the system of equations using Gaussian elimination method.

The system of equations is:

1. x - y + z = 4
2. 5x - y + z = 6
3. 3x - y + 5z = 5

Step 1: Write the system of equations in matrix form. The augmented matrix is:

[ 1 -1  1 | 4 ]
[ 5 -1  1 | 6 ]
[ 3 -1  5 | 5 ]

Step 2: Subtract 5 times the first row from the second row and 3 times the first row from the third row to eliminate x from the second and third equations. The new matrix is:

[ 1 -1  1 | 4 ]
[ 0  4  0 | -14 ]
[ 0  2  2 | -7 ]

Step 3: Divide the second row by 4 and subtract the second row from the third row to eliminate y from the third equation. The new matrix is:

[ 1 -1  1 | 4 ]
[ 0  1  0 | -3.5 ]
[ 0  0  2 | 0 ]

Step 4: Divide the third row by 2 to make the coefficient of z equal to 1. The new matrix is:

[ 1 -1  1 | 4 ]
[ 0  1  0 | -3.5 ]
[ 0  0  1 | 0 ]

Step 5: Add the first row to the second row and subtract the third row from the first row to eliminate z from the first and second equations. The new matrix is:

[ 1  0  0 | 4 ]
[ 0  1  0 | 0.5 ]
[ 0  0  1 | 0 ]

The solution to the system of equations is x = 4, y = 0.5, and z = 0.