Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

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

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{rcl}3x+3y+2z& =& 4\\ x-y-z& =& 0\end{array}\\ 2y-3z=-8\end{array}\right. \\ x=1, y=-1, z=2;\left(1,-1,2\right) \end{gathered}$$

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:

$\begin{array}{rcl}3x+3y+2z& =& 4\\ x-y-z& =& 0\\ 2y-3z& =& -8\end{array}$

The given solution is $x=1, y=-1, z=2$ or $\left(1,-1,2\right)$.

Step 2. Check the solution.

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

Substitute $x=1, y=-1, z=2$ in the first equation.

$\begin{array}{rcl}3\left(1\right)+3(-1)+2\left(2\right)& =& 3-3+4\\ & =& 4\end{array}$

So, the first equation is true.

Substitute $x=1, y=-1, z=2$ in the second equation.

$\begin{array}{rcl}1-(-1)-\left(2\right)& =& 1+1-2\\ & =& 0\end{array}$

So, the second equation is true.

Substitute $x=1, y=-1, z=2$ in the third equation.

$\begin{array}{rcl}2(-1)-3\left(2\right)& =& -2-6\\ & =& -8\end{array}$

So, the third equation is true.

Step 3. Conclusion.

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