Question

How to solve the following system of equations.

{ 2x-y+4z=33
x+2y-3z=-26
-5x-3y+5z=23}
solve using the gaussian elimination or a matrix. if you use a matrix you must write out your original matrix and answer matrix.

Answer 1

The student's system of equations can be solved using Gaussian elimination by transforming the original matrix into an upper triangular form and then using back substitution to find the values of x, y, and z.

Step-by-step explanation:

To solve the system of equations using Gaussian elimination, we first write the original matrix based on the coefficients and constants from the equations:

\[\begin{matrix} 2 & -1 & 4 & | & 33\\ 1 & 2 & -3 & | & -26\\ -5 & -3 & 5 & | & 23\end{matrix}\]

Gaussian elimination involves using row operations to transform the matrix into upper triangular form. This means we want zeros below the main diagonal of the matrix. We can then use back substitution to solve for the variables x, y, z. I'll outline the steps:

1. Use R2 - \(\frac{1}{2}\)R1 to make the coefficient of x in the second row zero.
2. Use R3 + 5R1 to make the coefficient of x in the third row zero.
3. Modify the second and third rows to achieve a zero beneath the main diagonal for y.
4. Back substitute starting from the third equation to find z.
5. Use z to solve for y using the second equation.
6. Use z and y to solve for x using the first equation.

The answer matrix will display the values of x, y, and z along the main diagonal with zeros below.