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: Perform row operations to get the matrix in row-echelon form. We can subtract 5 times the first row from the second row, and 3 times the first row from the third row:

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

Step 3: Now, let's subtract 2 times the second row from the third row:

[ 1 -1  1 | 4 ]
[ 0  4 -4 | -14 ]
[ 0  0  6 | 21 ]

Step 4: Divide the second row by 4 and the third row by 6 to make the leading coefficient in each row equal to 1:

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

Step 5: Now, let's perform back substitution. From the third equation, we have z = 3.5. Substituting z into the second equation gives y = -1. Substituting y and z into the first equation gives x = 2.

So, the solution to the system of equations is x = 2, y = -1, z = 3.5.