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Solve the following system

x+2y - 4z = -4
2x + 5y - 9z = -10
3x -2y + 3z = 11
by using Gauss Elimination

1 Answer

4 votes

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.

User Tempidope
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