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

$[A|B]=\left[\begin{array}{rrr|r} { 1 }&{-1} &{1} &{ 10}\\ {3 }&{3} &{0} &{ 5}\\ { 1 }&{1} &{2} &{2}\end{array}\right]$
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 have: $A=\left[\begin{array}{ll} 1 & -1 & 1\\ 3 & 3 &0 \\ 1&2&2 \end{array}\right]$ and $B=\left[\begin{array}{l} 10\\ 5\\2 \end{array}\right]$ Finally, we can write the system as an augmented matrix $[A|B]$ as follows: $[A|B]=\left[\begin{array}{rrr|r} { 1 }&{-1} &{1} &{ 10}\\ {3 }&{3} &{0} &{ 5}\\ { 1 }&{1} &{2} &{2}\end{array}\right]$