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 $\varnothing$.

Question

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 $\varnothing$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start with the system of equations: $ x + 2y + 3z = 1 $, $ -x + 3y + 5z = -5 $, and $ x + y - 3z = -7 $. ‖ ‖ 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.⚹◊What steps would you take to determine if the given system of linear equations has a unique solution, no solution, or infinitely many solutions using Gaussian elimination⍭ Generate me a similar question◊

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