Q. 67 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.  

$\left\{\begin{array}{l}\begin{array}{c}x-y+z=5\\ 3x+2y-2z=0\end{array}\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the given system of equations is

$x=2 , y=z-3$ ; where $z$ is any real number.

Step 1: Given information

The given system of equations is

$\left\{\begin{array}{l}\begin{array}{c}x-y+z=5\\ 3x+2y-2z=0\end{array}\end{array}\right.$

Step 2 : Concept used

Row operation :

1.  Interchange any two rows.
2.  Replace a row by a nonzero multiple of that row.
3.  Replace a row by the sum of that row and a constant nonzero multiple of some other row.

Step 3 : Calculation

The augmented matrix of the system is:    

$\left[\begin{array}{cc}\begin{array}{ccc}1& -1& 1\\ 3& 2& -2\end{array}& \begin{array}{c}\begin{array}{c}5\\ 0\end{array}\end{array}\end{array}\right]$

Perform the row operations $R_2=r_2-3r_1$:

$\left[\begin{array}{cc}\begin{array}{ccc}1& -1& 1\\ 0& 5& -5\end{array}& \begin{array}{c}\begin{array}{c}5\\ -15\end{array}\end{array}\end{array}\right]$

Perform the row operations $R_2=\frac{r_2}{5}$:

$\left[\begin{array}{cc}\begin{array}{ccc}1& -1& 1\\ 0& 1& -1\end{array}& \begin{array}{c}\begin{array}{c}5\\ -3\end{array}\end{array}\end{array}\right]$

Perform the row operations $R_1=r_1+r_2$ :

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

This matrix is in row echelon form.

Use the obtained matrix to write the system of equations.

$x=2 ...\left(1\right)$

$y-z=-3 ...\left(2\right)$

Step 4: Solve the equation.

From equation (2), 

$y=z-3$

Therefore, the solution of the system of equations is

$x=2 , y=z-3$; where $z$ is any real number.

Step 5 : Conclusion

The solution of the given system of equations is

$x=2 , y=z-3$ ; where $z$ is any real number.