To solve the system of equations using Gauss-Jordan elimination, we'll start by writing the augmented matrix for the system. The augmented matrix is formed by combining the coefficients of the variables and the constant terms on the right side of each equation:
[1 -4 1 | 0]
[2 2 -1 | -4]
[1 -2 -1 | -5]
Now, we'll apply row operations to transform the augmented matrix into reduced row-echelon form.
Let's perform row 2 - 2 * row 1 to eliminate the x term in the second row:
[1 -4 1 | 0]
[0 10 -3 | -4]
[1 -2 -1 | -5]
Next, perform row 3 - row 1 to eliminate the x term in the third row:
[1 -4 1 | 0]
[0 10 -3 | -4]
[0 2 -2 | -5]
To make the second element of the third row equal to zero, perform row 3 - (1/5) * row 2:
[1 -4 1 | 0]
[0 10 -3 | -4]
[0 0 -1 | -3/5]
We can multiply the third row by -1 to make the leading coefficient in the third row positive:
[1 -4 1 | 0]
[0 10 -3 | -4]
[0 0 1 | 3/5]
Now, let's perform row 2 - 3 * row 3 to eliminate the z term in the second row:
[1 -4 1 | 0]
[0 10 0 | -19/5]
[0 0 1 | 3/5]
Next, perform row 1 + 4 * row 3 to eliminate the z term in the first row:
[1 -4 0 | 12/5]
[0 10 0 | -19/5]
[0 0 1 | 3/5]
Finally, divide the second row by 10 and simplify:
[1 -4 0 | 12/5]
[0 1 0 | -19/50]
[0 0 1 | 3/5]
Divide the first row by -4 and simplify:
[-1/4 1 0 | -3/5]
[0 1 0 | -19/50]
[0 0 1 | 3/5]
The resulting matrix corresponds to the system:
-1/4x + y = -3/5
y = -19/50
z = 3/5
Therefore, the solution to the system of equations is:
x = -3/10
y = -19/50
z = 3/5