Task 2:  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

First, let's write this system in augmented matrix form:

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

Next, we'll perform row operations to get this matrix in row-echelon form.

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 ]

Now, divide the second row by 4 and the third row by 2:

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

Subtract the second row from the third row:

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

Divide the third row by 2:

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

Now, we can use back substitution to solve for the variables.

From the third row, we have z = 0.

Substitute z = 0 into the second row, we get y = -3.5.

Substitute y = -3.5 and z = 0 into the first row, we get x = 1.5.

So, the solution to the system of equations is x = 1.5, y = -3.5, and z = 0.