## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$[A|B]=\left[\begin{array}{rrr|r} {5 }&{-1} &{-1} &{0 }\\ {1 }&{1}&{0} &{5 }\\{2}&{0}&{-3}&{2}\end{array}\right]$
The standard form of a linear equation can be expressed as: $$a_{i1}x_{1}+a_{i2}x_{2}+...+a_{in}x_{n}=b_{i}$$ where the index $i$ indicates that it is the i-th equation of a system of equations. In order to write the augmented matrix $[A|B]$ of a system of equations in standard form, we must follow some important points: 1. To express a system in matrix form, we must extract the coefficients of the variables and constants. 2. Draw a vertical line to separate the coefficient entries from the constants (essentially replacing the equal signs). 3. The entries of the coefficient matrix $A=[a_{ij}]$ must be placed to the left of the lline. 4. The constants of the $B=[b_{i}]$ must be placed to the right of the line. We rewrite the system in standard form as: $5x-y-z=0 \\x+y=5 \\2x-3z=2$ Now, we have: $A=\left[\begin{array}{ll} 5 & -1 &-1\\ 1 & 1&0 \\2&0&-3 \end{array}\right]$ and $B=\left[\begin{array}{l} 0\\ 5\\2 \end{array}\right]$ Finally, we can write the system as an augmented matrix $[A|B]$ as follows: $[A|B]=\left[\begin{array}{rrr|r} {5 }&{-1} &{-1} &{0 }\\ {1 }&{1}&{0} &{5 }\\{2}&{0}&{-3}&{2}\end{array}\right]$