Final answer:
Using Gauss Elimination, we transform the given system of equations into an upper triangular matrix using row operations, then use back substitution to solve for x, y, and z.
Step-by-step explanation:
To solve the system of equations using Gauss Elimination, let's write down the augmented matrix representing the system:
1 2 -4 | -4
2 5 -9 | -10
3 -2 3 | 11
We aim to transform this matrix into an upper triangular form using elementary row operations. First, let's eliminate the x terms from the second and third equations by making them zeros below the leading 1 of the first row:
Multiply the first row by 2 and subtract it from the second row.
Multiply the first row by 3 and subtract it from the third row.
1 2 -4 | -4
0 1 -1 | -2
0 -8 15 | 23
Next, we make the y term in the third equation zero:
Multiply the second row by 8 and add it to the third row.
1 2 -4 | -4
0 1 -1 | -2
0 0 3 | 7
Now, we have an upper triangular matrix, and we can use back substitution to find the values of z, y, and x:
The third equation is now 3z = 7, so z = 7/3.
Substitute the value of z into the second equation to find y.
Finally, substitute the values of y and z into the first equation to find x.
After solving, we get the solutions for x, y, and z.