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

The system of equations is: $\quad \quad \begin{array}{llll} &\quad x &- &3y &+ &3z & =-5\\ &-4x &- &5y &- &3z & =-5\\ &-3x &- &2y &+& 4z & = \space 6 \end{array}$ Performing theh indicated operations yield: $\quad \left[\begin{array}{ccc|c} {1}&{-3}&{3}&{-5}\\ {0}&{-17}&{9}&{-25}\\ {0}&{-11}&{13}&{-9}\end{array}\right]$
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 can write the system as an augmented matrix $[A|B]$ as follows: $\left\{\begin{array}{llll} x & -3y & +3z & =-5\\ -4x & -5y & -3z & =-5\\ -3x & -2y & +4z & =6 \end{array}\right. \rightarrow\left[\begin{array}{rrr|r} {1}&{-3}&{3}&{-5}\\ {-4}&{-5}&{-3}&{-5}\\ {-3}&{-2}&{4}&{6}\end{array}\right]$ We perform the row operations as: $R_{2}=4r_{1}+r_{2}$ $R_{3}=3r_{1}+r_{3}$ $=\left[\begin{array}{ccc|c} {1} &{-3} &{3} &{-5}\\ {4(1)-4}&{4(-3)-5}&{4(3)-3}&{4(-5)-5}\\ {3(1)-3}&{3(-3)-2}&{3(3)+4}&{3(-5)+6}\end{array}\right]$ $=\left[\begin{array}{ccc|c} {1}&{-3}&{3}&{-5}\\ {0}&{-17}&{9}&{-25}\\ {0}&{-11}&{13}&{-9}\end{array}\right]$