Question

To determine whether the ordered triple is a solution to the system.

$\left\{\begin{array}{l}-3x+y+z=-4\\ -x+2y-2z=1\\ 2x-y-z=-1\end{array}\right.$

localid="1644429233421" $$\begin{gathered} \left(a\right)(-5,-7,4) \\ \left(b\right)(5,7,4) \end{gathered}$$

Short answer

The ordered triple (-5,-7,4) ls not a solution of the system of equations.

The ordered triple (5,7,4) is a solution of the system of linear equations.

Step-by-step solution

Part (a) Step 1. Given information

Given system of linear equations:

$$\begin{gathered} -3x+y+z=-4 \\ -x+2y-2z=1 \\ 2x-y-z=-1 \end{gathered}$$

And coordinates:$\left(a\right)(-5,-7,4)$

Part (a) Step 2. Testing

First substitute:

$$\begin{gathered} x=-5 \\ y=-7 \\ z=4 \end{gathered}$$in equation one we get:

$\begin{array}{r}-3x+y+z=-4\\ -3(-5)-7+4=-4\\ 15-7+4=-4\\ 12=-4\end{array}$

This is not true.

So $(-5,-7,4)$is not a solution.

Part (b) Step 1. Given information

Given equation:$$\begin{gathered} -3x+y+z=-4 \\ -x+2y-2z=1 \\ 2x-y-z=-1 \end{gathered}$$

and coordinates:

$(5,7,4)$

Part (b) Step 2. Solving

Substituting given value of

$$\begin{gathered} x=5, \\ y=7, \\ z=4 \end{gathered}$$

in the equation one.

$\begin{array}{r}-3x+y+z=-4\\ -3\left(5\right)+7+4=-4\\ -15+7+4=-4\\ -4=-4\end{array}$

This is true.

Also substitute the same values in equation two:

$$\begin{gathered} -x+2y-2z=1 \\ -\left(5\right)+2\left(7\right)-2\left(4\right)=1 \\ -5+14-8=1 \\ 1=1 \end{gathered}$$

This is true.

Subtitling the given coordinates in equation three.

$$\begin{gathered} 2x-y-z=-1 \\ 2\left(5\right)-\left(7\right)-4=-1 \\ 10-7-4=-1 \\ -1=-1 \end{gathered}$$

This is true.

So$(5,7,4)$is solution.