Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable.
$\begin{array} { l }
x - z = - 5 \
y + z = 1 \
x + z = 2
\end{array}$
A) $\{ ( - 3 , - 5,7 ) \}$
B) $\left\{ \left( \frac { 7 } { 2 } , - \frac { 15 } { 2 } , \frac { 17 } { 2 } \right) \right\}$
C) $\{ ( - 5,1,0 ) \}$
D) $\left\{ \left( - \frac { 3 } { 2 } , - \frac { 5 } { 2 } , \frac { 7 } { 2 } \right) \right\}$

Question

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable.
$\begin{array} { l } 
x - z = - 5 \
y + z = 1 \
x + z = 2
\end{array}$
A) $\{ ( - 3 , - 5,7 ) \}$
B) $\left\{ \left( \frac { 7 } { 2 } , - \frac { 15 } { 2 } , \frac { 17 } { 2 } \right) \right\}$
C) $\{ ( - 5,1,0 ) \}$
D) $\left\{ \left( - \frac { 3 } { 2 } , - \frac { 5 } { 2 } , \frac { 7 } { 2 } \right) \right\}$

Quizplus · Accepted Answer

To solve the system using the Gauss-Jordan method, we first write the augmented matrix and then apply row operations to get it in reduced row echelon form (RREF). The system of equations is:1. $x - z = -5$2. $y + z = 1$3. $x + z = 2$The augmented matrix is:$$\left[\begin{array}{ccc|c}1 & 0 & -1 & -5 \0 & 1 & 1 & 1 \1 & 0 & 1 & 2 \\end{array}\right]$$Applying row operations to get it in RREF, we aim for a matrix that looks like this:$$\left[\begin{array}{ccc|c}1 & 0 & 0 & a \0 & 1 & 0 & b \0 & 0 & 1 & c \\end{array}\right]$$Row operations:1. R3 = R3 - R1, resulting in:$$\left[\begin{array}{ccc|c}1 & 0 & -1 & -5 \0 & 1 & 1 & 1 \0 & 0 & 2 & 7 \\end{array}\right]$$2. R3 = R3 / 2, resulting in:$$\left[\begin{array}{ccc|c}1 & 0 & -1 & -5 \0 & 1 & 1 & 1 \0 & 0 & 1 & \frac{7}{2} \\end{array}\right]$$3. R1 = R1 + R3, R2 = R2 - R3, resulting in:$$\left[\begin{array}{ccc|c}1 & 0 & 0 & -\frac{3}{2} \0 & 1 & 0 & -\frac{5}{2} \0 & 0 & 1 & \frac{7}{2} \\end{array}\right]$$This corresponds to the solution $x = -\frac{3}{2}$, $y = -\frac{5}{2}$, and $z = \frac{7}{2}$, which matches option D.

Use the Gauss-Jordan Method to Solve the System of Equations x−z=−5y+z=1x+z=2\begin{array} { l } x - z = - 5 \\y + z = 1 \\x + z = 2\end{array}x−z=−5y+z=1x+z=2​

Use the Gauss-Jordan Method to Solve the System of Equations $\begin{array} { l } x - z = - 5 \\y + z = 1 \\x + z = 2\end{array}$