Q. 16 - Precalculus Enhanced with Graphing Utilities

Question

Verify that the values of the variables listed are solutions of the system of equations.

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{rcl}4x-5z& =& 6\\ 5y-z& =& -17\end{array}\\ -x-6y+5z=24\end{array}\right. \\ x=4, y=-3, z=2; \left(4,-3,2\right) \end{gathered}$$

Step-by-Step Solution

Answer

The given values are the solutions because all equations are true for the given values.

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}4x-5z& =& 6\\ 5y-z& =& -17\\ -x-6y+5z& =& 24\end{array}$

The given solution is $x=4, y=-3, z=2$ or $\left(4,-3,2\right)$.

Step 2. Check the solution.

If all equations are true for the given values, then the values represent the solution.

Substitute $x=4, y=-3, z=2$ in the first equation.

$\begin{array}{rcl}4\left(4\right)-5\left(2\right)& =& 16-10\\ & =& 6\end{array}$

So, the first equation is true.

Substitute $x=4, y=-3, z=2$ in the second equation.

$\begin{array}{rcl}5(-3)-\left(2\right)& =& -15-2\\ & =& -17\end{array}$

So, the second equation is true.

Substitute  in the third equation.

$\begin{array}{rcl}-\left(4\right)-6(-3)+5\left(2\right)& =& -4+18+10\\ & =& 24\end{array}$

So, the third equation is true.

Step 3. Conclusion.

The given values are the solutions because all equations are true for the given values.