Problem 1. (10 points)
Linear System - Three Variables
Solve the following system of equations using Gaussian elimination method. If there are no solutions, type "N" for both $x, y$ and $z$. If there are infinitely many solutions, type " $z$ " for $z$, and expressions in terms of $z$ for $x$ and $y$.
$$
\begin{aligned}
4 x-3 y-5 z & =-4 \
3 x+1 y-8 z & =-1 \
-5 x+1 y+6 z & =-3
\end{aligned}
$$
$x=$ $\qquad$ $y=$ $\qquad$ $z=$ $\qquad$
Answer(s) submitted:

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Write the augmented matrix for the given system of equations: $$ \begin{pmatrix} 4 & -3 & -5 & | & -4 \ 3 & 1 & -8 & | & -1 \ -5 & 1 & 6 & | & -3 \end{pmatrix} $$ ‖ ‖ step 2 ⋮ Perform row operations to get a leading 1 in the first row, first column. Divide the first row by 4: $$ \begin{pmatrix} 1 & -\frac{3}{4} & -\frac{5}{4} & | & -1 \ 3 & 1 & -8 & | & -1 \ -5 & 1 & 6 & | & -3 \end{pmatrix} $$ ‖ ‖ step 3 ⋮ Eliminate the first column entries below the leading 1 by performing row operations: $$ R2 = R2 - 3R1 \quad \text{and} \quad R3 = R3 + 5R1 $$ $$ \begin{pmatrix} 1 & -\frac{3}{4} & -\frac{5}{4} & | & -1 \ 0 & \frac{13}{4} & -\frac{17}{4} & | & 2 \ 0 & -\frac{11}{4} & \frac{11}{4} & | & -8 \end{pmatrix} $$ ‖ ‖ step 4 ⋮ Normalize the second row by dividing by $\frac{13}{4}$: $$ R2 = \frac{4}{13} R2 $$ $$ \begin{pmatrix} 1 & -\frac{3}{4} & -\frac{5}{4} & | & -1 \ 0 & 1 & -\frac{17}{13} & | & \frac{8}{13} \ 0 & -\frac{11}{4} & \frac{11}{4} & | & -8 \end{pmatrix} $$ ‖ ‖ step 5 ⋮ Eliminate the second column entries below and above the leading 1 in the second row: $$ R3 = R3 + \frac{11}{4} R2 \quad \text{and} \quad R1 = R1 + \frac{3}{4} R2 $$ $$ \begin{pmatrix} 1 & 0 & -\frac{29}{13} & | & -\frac{5}{13} \ 0 & 1 & -\frac{17}{13} & | & \frac{8}{13} \ 0 & 0 & 0 & | & -\frac{10}{13} \end{pmatrix} $$ ‖ ‖ step 6 ⋮ Since the third row results in $0 = -\frac{10}{13}$, which is a contradiction, the system has no solutions. ‖ ∻Answer∻ ⚹ $x = N$, $y = N$, $z = N$ ⚹ ∻Key Concept∻ ⚹Gaussian Elimination⚹ ∻Explanation∻ ⚹Gaussian elimination is used to solve systems of linear equations by transforming the system's augmented matrix into row-echelon form. If a row results in a contradiction, the system has no solutions.⚹◊What is the next step in solving the system of equations using the Gaussian elimination method?⍭ Generate me a similar question◊

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