Question 8.1.1: Writing an Augmented Matrix Write the augmented matrix for t...
Writing an Augmented Matrix
Write the augmented matrix for the system of linear equations:
\left\{\begin{aligned}2 y-z &=7 \\x+2 y+z &=17 \\2 x-3 y+2 z &=-1.\end{aligned}\right.
Begin by showing the coefficient of each variable. Replace the missing x-variable in the first equation with 0x. For clarity, we’ve also numbered the equations.
\begin{matrix} \text{Equation 1} \\ \text{Equation 2} \\ \text{Equation 3} \end{matrix} \left\{\begin{aligned}2 y-z &=7 \\x+2 y+z &=17 \\2 x-3 y+2 z &=-1\end{aligned}\right.\begin{matrix} \longrightarrow \\ \longrightarrow \\ \longrightarrow \end{matrix}\left\{\begin{array}{l}0 x+2 y-1 z=7 \\1 x+2 y+1 z=17 \\2 x-3 y+2 z=-1\end{array}\right. \begin{matrix} \text{Use this equation to obtain row 1 of the augmented matrix.} \\ \text{Use this equation to obtain row 2 of the augmented matrix.} \\ \text{Use this equation to obtain row 3 of the augmented matrix.} \end{matrix}Now we are ready to write the augmented matrix.
We used Equation 1 to obtain the numbers in row 1 . The coefficients of each variable, 0,2 , and -1, are placed to the left of the vertical line in the augmented matrix. The constant, 7 , is placed to the right of the vertical line. In a similar way, we used Equation 2 to obtain the numbers in row 2. We used Equation 3 to obtain the numbers in row 3 .
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