Use Gaussian elimination to solve the system of equations.
-x - y + w - 3z = 3 x + 2y = -3
Y + w = -2
-x + w + z = 3
A) (26, -15, 13, 17)
B) (25, -14, 12, 16)
C) (23, -13, 11, 17)
D) (27, -14, 12, 17)

Question

Quizplus · Accepted Answer

To solve the system of equations using Gaussian elimination, we first write the equations in augmented matrix form and then perform row operations to get it into reduced row echelon form (RREF). The system of equations is:1. $x - y + w - 3z = 3$2. $x + 2y = -3$3. $y + w = -2$4. $-x + w + z = 3$In matrix form, this is:$$\begin{align*}\begin{bmatrix}1 & -1 & 1 & -3 & | & 3 \1 & 2 & 0 & 0 & | & -3 \0 & 1 & 1 & 0 & | & -2 \-1 & 0 & 1 & 1 & | & 3 \\end{bmatrix}\end{align*}$$After performing Gaussian elimination (which involves a series of row operations to simplify the matrix), we aim to get the matrix into a form where we can easily read off the solutions for $x$, $y$, $w$, and $z$. The exact steps of the elimination process are not detailed here, but the end result gives us the solution for the variables.The correct solution, after performing Gaussian elimination and back substitution, is $x = 25$, $y = -14$, $w = 12$, and $z = 16$, which corresponds to choice B. This process involves a series of steps to simplify the matrix and solve for each variable, starting from the bottom row and working upwards, substituting known values as you go.

Use Gaussian Elimination to Solve the System of Equations