Use Matrices and Gaussian Elimination to Solve Systems
-$$\begin{aligned}
5 x - y - 3 z & = - 8 \
3 x - 8 z & = - 21 \
7 y + z & = 31
\end{aligned}$$
A) $\{ ( 1,4,3 ) \}$
B) $\{ ( 1,3,4 ) \}$
C) $\{ ( - 1,4,2 ) \}$
D) $\{ ( - 1,2,4 ) \}$

Question

Quizplus · Accepted Answer

To solve the system using matrices and Gaussian elimination, we first write the augmented matrix for the system, then apply row operations to reach reduced row echelon form (RREF). The system is:$$\begin{align*}5x - y - 3z &= -8 \3x + 0y - 8z &= -21 \0x + 7y + z &= 31\end{align*}$$The augmented matrix is:$$\begin{bmatrix}5 & -1 & -3 & | & -8 \3 & 0 & -8 & | & -21 \0 & 7 & 1 & | & 31 \\end{bmatrix}$$Applying Gaussian elimination steps (which are not detailed here due to the brief explanation requirement), we aim to transform this matrix into RREF. The solution obtained from the RREF will match option A, which is $\{ (1,4,3) \}$, indicating that $x=1$, $y=4$, and $z=3$ are the solutions to the system.

Use Matrices and Gaussian Elimination to Solve Systems - 5x−y−3z=−83x−8z=−217y+z=31\begin{aligned}5 x - y - 3 z & = - 8 \\3 x - 8 z & = - 21 \\7 y + z & = 31\end{aligned}5x−y−3z3x−8z7y+z​=−8=−21=31​

Use Matrices and Gaussian Elimination to Solve Systems
- $\begin{aligned}5 x - y - 3 z & = - 8 \\3 x - 8 z & = - 21 \\7 y + z & = 31\end{aligned}$