Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\{ \begin{aligned}
x - 3 z & = - 7 \
3 x + y - 2 z & = 2 \
2 x + 2 y + z & = 4
\end{aligned} \right.$$
A)(-5,-5,4)
B)(5,-5,-4)
C)(-5,-5,-4)
D)(5,5,4)
E)(5,-5,4)

Question

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $$\left\{ \begin{aligned}
x - 3 z & = - 7 \
3 x + y - 2 z & = 2 \
2 x + 2 y + z & = 4
\end{aligned} \right.$$ ​
A)(-5,-5,4)
B)​(5,-5,-4)
C)​(-5,-5,-4)
D)​(5,5,4)
E)​(5,-5,4)

Quizplus · Accepted Answer

To solve the system of equations using matrices and Gaussian elimination, we first write the augmented matrix corresponding to the system:$$\begin{align*}\begin{bmatrix}1 & 0 & -3 & | & -7 \3 & 1 & -2 & | & 2 \2 & 2 & 1 & | & 4\end{bmatrix}\end{align*}$$We aim to transform this matrix into row-echelon form (REF) or reduced row-echelon form (RREF) through a series of row operations.1. Subtract 3 times the first row from the second row to eliminate the $x$ term in the second equation.2. Subtract 2 times the first row from the third row to eliminate the $x$ term in the third equation.3. Make the leading coefficient of the second row 1 (if it's not already).4. Use the second row to eliminate the $y$ term from the third row.5. Back substitute if necessary to solve for $x$, $y$, and $z$.After performing these operations, we aim to get a matrix that allows us to directly read off the values of $x$, $y$, and $z$. The exact operations will depend on the specifics of the matrix as we proceed, but the goal is to get to a point where the matrix looks something like this (in RREF):$$\begin{align*}\begin{bmatrix}1 & 0 & 0 & | & a \0 & 1 & 0 & | & b \0 & 0 & 1 & | & c\end{bmatrix}\end{align*}$$Where $a$, $b$, and $c$ are the solutions for $x$, $y$, and $z$ respectively.Performing the operations, we find that the solution to the system is $x = 5$, $y = -5$, and $z = 4$, which corresponds to choice E) $(5,-5,4)$.

Use Matrices to Solve the System of Equations (If Possible) {x−3z=−73x+y−2z=22x+2y+z=4\left\{ \begin{aligned}x - 3 z & = - 7 \\3 x + y - 2 z & = 2 \\2 x + 2 y + z & = 4\end{aligned} \right.⎩⎨⎧​x−3z3x+y−2z2x+2y+z​=−7=2=4​

Use Matrices to Solve the System of Equations (If Possible) $\left\{ \begin{aligned}x - 3 z & = - 7 \\3 x + y - 2 z & = 2 \\2 x + 2 y + z & = 4\end{aligned} \right.$