Q 4.62 - Intermediate Algebra

Question

Determine whether the ordered triple is a solution to the system:

$\left\{\begin{array}{l}x-3y+z=-5\\ -3x-y-z=1\\ 2x-2y+3z=1\end{array}\right.$

(a) $(2,-2,3)$

(b) $(-2,2,3)$

Step-by-Step Solution

Answer

Part (a) The given ordered triple $(2,-2,3)$ is not a solution of the system of linear equations.

Part (a) The given ordered triple $(-2,2,3)$ is a solution of the given system of linear equations. 

Part (a) Step 1. Given information

We are given system of linear equations $\left\{\begin{array}{l}x-3y+z=-5\\ -3x-y-z=1\\ 2x-2y+3z=1\end{array}\right.$ and coordinate $(2,-2,3)$.

Part (a) Step 2 . Testing

Substituting $$\begin{gathered} x=2 \\ y=-2 \\ z=3 \end{gathered}$$ in given equation and checking whether the equations is true or not.

Checking first equation $x-3y+z=-5$,

$$\begin{gathered} 2-3(-2)+3=-5 \\ 2+6+3=-5 \\ 11\ne -5 \end{gathered}$$

Hence $(2,-2,3)$ is not the solution of given system of linear equation.

Part (b) Step 1. Given information

We are given system of linear equations $\left\{\begin{array}{l}x-3y+z=-5\\ -3x-y-z=1\\ 2x-2y+3z=1\end{array}\right.$and coordinate $(-2,2,3)$.

Part (b) Step 2. Testing

Substituting $$\begin{gathered} x=-2 \\ y=2 \\ z=3 \end{gathered}$$ in given equation and checking whether the equations is true or not.

Checking first equation $x-3y+z=-5$,

$$\begin{gathered} -2-3\left(2\right)+3=-5 \\ -2-6+3=-5 \\ -5=-5 \end{gathered}$$

This is True.

Checking second equation $-3x-y-z=1$,

$$\begin{gathered} -3(-2)-2-3=1 \\ 6-2-3=1 \\ 1=1 \end{gathered}$$

This is also true.

Checking third equation $2x-2y+3z=1$,

$$\begin{gathered} 2(-2)-2\left(2\right)+3\left(3\right)=1 \\ -4-4+9=1 \\ 1=1 \end{gathered}$$

This is also true.

Hence $(-2,2,3)$ is solution of given system of linear equations.