Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{array} { l }
5 x - y + z = 8 \
7 x + y + z = 6
\end{array}$$
A) $\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}$
B) $\{ ( - \mathrm { z } + 3,4 \mathrm { z } + 7 , \mathrm { z } ) \}$
C) $\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}$
D) $\varnothing$

Question

Apply Gaussian Elimination to Systems with More Variables than Equations
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
-$$\begin{array} { l } 
5 x - y + z = 8 \
7 x + y + z = 6
\end{array}$$
A) $\left\{ \left( - \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z - \frac { 13 } { 6 } , z \right) \right\}$
B) $\{ ( - \mathrm { z } + 3,4 \mathrm { z } + 7 , \mathrm { z } ) \}$
C) $\left\{ \left( \frac { 1 } { 6 } z + \frac { 7 } { 6 } , \frac { 1 } { 6 } z , z \right) \right\}$
D) $\varnothing$

Quizplus · Accepted Answer

We can write the augmented matrix for the system as:

\begin{bmatrix}5 & -1 & 1 & 8\7 & 1 & 1 & 6\end{bmatrix}

We apply Gaussian elimination to obtain:

\begin{bmatrix}5 & -1 & 1 & 8\0 & 8 & -2 & -26\end{bmatrix}

The reduced row echelon form of this matrix is:

\begin{bmatrix}1 & 0 & \frac{1}{4} & \frac{7}{4}\0 & 1 & -\frac{1}{4} & -\frac{13}{4}\end{bmatrix}

From this, we can see that:

$$x=\frac{1}{4}z+\frac{7}{4}$$
$$y=-\frac{1}{4}z-\frac{13}{4}$$

We can then write the solution as:

$$\left\{\left(-\frac{1}{6}z+\frac{7}{6},\frac{1}{6}z-\frac{13}{6},zight)ight\}$$

Apply Gaussian Elimination to Systems with More Variables Than Equations 5x−y+z=87x+y+z=6\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}5x−y+z=87x+y+z=6​

Apply Gaussian Elimination to Systems with More Variables Than Equations $\begin{array} { l } 5 x - y + z = 8 \\7 x + y + z = 6\end{array}$