Use the Gauss-Jordan method to solve the system of equations.
-$$
\begin{array}{l}
5 x-y+z=8 \
7 x+y+z=6 \
12 x+2 z=14
\end{array}
$$
A) $ \left(\frac{-z+7}{6}, \frac{z-13}{6}, z\right) $
B) $ \left(\frac{z+7}{6}, \frac{-z-13}{6}, z\right) $
C) $ \left(\frac{-z+7}{6}, \frac{z+13}{6}, z\right) $
D) $ \left(\frac{z-7}{6}, \frac{-z-13}{6}, z\right) $

Question

Quizplus · Accepted Answer

To solve the system using the Gauss-Jordan method, we first write the augmented matrix of the system and then apply row operations to get it in reduced row echelon form (RREF). The system of equations is:$$\begin{align*}5x - y + z &= 8 \7x + y + z &= 6 \12x + 0y + 2z &= 14\end{align*}$$The augmented matrix is:$$\begin{bmatrix}5 & -1 & 1 & | & 8 \7 & 1 & 1 & | & 6 \12 & 0 & 2 & | & 14\end{bmatrix}$$Applying row operations to simplify this matrix into RREF, we aim to get leading ones in each row and zeros everywhere else in the column of those leading ones, except in the last column which represents the constants from the right-hand side of the equations.After performing the necessary row operations (which are too lengthy to detail in this brief explanation but involve making the leading coefficient of each row a 1, eliminating other numbers in that column to 0, and making sure each leading 1 moves to the right as we move down the rows), we find that the system of equations has one free variable (let's choose $z$ as the free variable since it's common to express the solution in terms of the last variable when possible).The solution in terms of $z$ matches option A: $ \left(\frac{-z+7}{6}, \frac{z-13}{6}, z\right) $. This indicates that $x$ and $y$ can be expressed in terms of $z$, and their specific relationships to $z$ are given in option A, which is derived from the final form of the augmented matrix after applying Gauss-Jordan elimination.

Use the Gauss-Jordan Method to Solve the System of Equations 5x−y+z=87x+y+z=612x+2z=14\begin{array}{l}5 x-y+z=8 \\7 x+y+z=6 \\12 x+2 z=14\end{array}5x−y+z=87x+y+z=612x+2z=14​

Use the Gauss-Jordan Method to Solve the System of Equations $\begin{array}{l}5 x-y+z=8 \\7 x+y+z=6 \\12 x+2 z=14\end{array}$