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Use the Gaussian elimination method to solve the system of linear equations. If the system has infinitely many solutions, write the solution set with z arbitrary.
$\begin{array}{rr}
x+2 y+3 z= & 1 \\
-x+3 y+5 z= & -5 \\
x+y-3 z= & -7
\end{array}$
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. There is one solution. The solution is $\{(\square, \square)\}$. $\square$
(Simplify your answers. Type integers or decimals.)
B. There are infinitely many solutions. The solutions are $\{(\square, z)\}$, where $z$ is any real number. $\square$ $\square$ (Simplify your answers. Type integers or decimals.)
C. The system is inconsistent. The solution set is $\varnothing$.
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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 2

We can represent the system in augmented matrix form: $\begin{bmatrix} 1 & 2 & 3 & | & 1 \\ -1 & 3 & 5 & | & -5 \\ 1 & 1 & -3 & | & -7 \end{bmatrix}$

step 3

We perform row operations to reach reduced row echelon form. First, we can replace Row 2 with Row 2 + Row 1: $\begin{bmatrix} 1 & 2 & 3 & | & 1 \\ 0 & 5 & 8 & | & -4 \\ 1 & 1 & -3 & | & -7 \end{bmatrix}$

step 4

Next, we replace Row 3 with Row 3 - Row 1: $\begin{bmatrix} 1 & 2 & 3 & | & 1 \\ 0 & 5 & 8 & | & -4 \\ 0 & -1 & -6 & | & -8 \end{bmatrix}$

step 5

We can simplify Row 2 and Row 3 further to isolate variables, leading to the conclusion that the system has infinitely many solutions expressed as: $\{(x, y, z) | z \text{ is arbitrary}\}$

B

Key Concept

Gaussian elimination and row operations

Explanation

This method allows us to systematically solve systems of linear equations by transforming the augmented matrix into a simpler form, revealing the relationships between the variables.

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