Question

Write the system of equations that corresponds to the augmented matrix.

$\left[\begin{array}{rrrr}1& 0& -3& -1\\ 1& -2& 0& -2\\ 0& -1& 2& 3\end{array}\right]$

Short answer

For the augmented matrix $\left[\begin{array}{rrrr}1& 0& -3& -1\\ 1& -2& 0& -2\\ 0& -1& 2& 3\end{array}\right]$, the system of equations is$\left\{\begin{array}{l}x-3z=-1\\ x-2y=-2\\ -y+2z=3\end{array}\right.$

Step-by-step solution

Step 1. Given Information

Consider the augmented matrix $\left[\begin{array}{rrrr}1& 0& -3& -1\\ 1& -2& 0& -2\\ 0& -1& 2& 3\end{array}\right]$.

We need to find the corresponding system of equations.

Step 2. Form the system of equations

We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a $4\times 3$, we know it will translate into a system of three equations with three variables. Thus the corresponding system of equations is

$$\begin{gathered} \left[\begin{array}{rrrr}1& 0& -3& -1\\ 1& -2& 0& -2\\ 0& -1& 2& 3\end{array}\right]\to \left\{\begin{array}{l}1·x+0·y+(-3)·z=-1\\ 1·x+(-2)·y+0·z=-2\\ 0·x+(-1)·y+2·z=3\end{array}\right. \\ \to \left\{\begin{array}{l}x-3z=-1\\ x-2y=-2\\ -y+2z=3\end{array}\right. \end{gathered}$$