Q. 163 - Intermediate Algebra

Question

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

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

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

Step-by-Step Solution

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.

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:

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

This is true.

Subtitling the given coordinates in equation three.

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

This is true.

So $(5,7,4)$ is solution.