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.
-$$\begin{array} { l }
x - z = - 2 \
y + z = 10 \
x + z = 3
\end{array}$$
A) $\left\{ \left\{ \frac { 5 } { 2 } , \frac { 11 } { 2 } , \frac { 9 } { 2 } \right) \right\}$

B) $\left\{ \left( \frac { 1 } { 2 } , \frac { 15 } { 2 } , \frac { 5 } { 2 } \right) \right\}$

C) $\{ ( - 2,10,0 ) \}$

D) $\{ ( 1,15,5 ) \}$

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.
-$$\begin{array} { l } 
x - z = - 2 \
y + z = 10 \
x + z = 3
\end{array}$$
A) $\left\{ \left\{ \frac { 5 } { 2 } , \frac { 11 } { 2 } , \frac { 9 } { 2 } \right) \right\}$

B) $\left\{ \left( \frac { 1 } { 2 } , \frac { 15 } { 2 } , \frac { 5 } { 2 } \right) \right\}$

C) $\{ ( - 2,10,0 ) \}$

D) $\{ ( 1,15,5 ) \}$

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 into reduced row echelon form (RREF). The system of equations is:1. $x - z = -2$2. $y + z = 10$3. $x + z = 3$The augmented matrix is:$$\begin{align*}\left[\begin{array}{ccc|c}1 & 0 & -1 & -2 \0 & 1 & 1 & 10 \1 & 0 & 1 & 3 \\end{array}\right]\end{align*}$$Applying row operations to get it into RREF, we aim to make the matrix look like this:$$\begin{align*}\left[\begin{array}{ccc|c}1 & 0 & 0 & a \0 & 1 & 0 & b \0 & 0 & 1 & c \\end{array}\right]\end{align*}$$By adding the first and third rows, we eliminate $z$ in those equations, and we can solve for $x$ and $y$ directly, considering $z$ as the arbitrary variable if necessary. However, in this case, we can solve for all variables explicitly:1. From $x - z = -2$ and $x + z = 3$, adding these gives $2x = 1$, so $x = \frac{1}{2}$.2. From $y + z = 10$ and knowing $x + z = 3$ gives $z = \frac{5}{2}$ by substituting $x$'s value.3. Substituting $z = \frac{5}{2}$ into $y + z = 10$ gives $y = \frac{15}{2}$.Thus, the solution is $\left( \frac{1}{2}, \frac{15}{2}, \frac{5}{2} \right)$, which matches choice B.

Use the Gauss-Jordan Method to Solve the System of Equations x−z=−2y+z=10x+z=3\begin{array} { l } x - z = - 2 \\y + z = 10 \\x + z = 3\end{array}x−z=−2y+z=10x+z=3​

Use the Gauss-Jordan Method to Solve the System of Equations $\begin{array} { l } x - z = - 2 \\y + z = 10 \\x + z = 3\end{array}$