Q. 368 - Intermediate Algebra

Question

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

$\left\{\begin{array}{l}3x-4y-3z=2\\ 2x-6y+z=3\\ 2x+3y-2z=3\end{array}\right.$

(a) $(2,3,-1)$

(b) $(3,1,3)$

Step-by-Step Solution

Answer

(a) The ordered triple $(2,3,-1)$ is not a solution to the system of linear equations.

(b) The ordered triple $(3,1,3)$ is not a solution of the system of linear equations.

Step 1. Given the information

The system of equations is,

$\left\{\begin{array}{l}3x-4y-3z=2..........\left(1\right)\\ 2x-6y+z=3...........\left(2\right)\\ 2x+3y-2z=3.........\left(3\right)\end{array}\right.$

Part a Step 1. Finding whether ( 2 , 3 , - 1 ) satisfies equation (1)

Substituting $$\begin{gathered} x=2 \\ y=3 \\ z=-1 \end{gathered}$$ in the equation

$3x-4y-3z=2$,

$$\begin{gathered} 3\left(2\right)-4\left(3\right)-3\left(1\right)=2 \\ 6-12-3=2 \\ 6-15=2 \\ -9\ne 2 \end{gathered}$$

Part a. Step 2. Finding whether ( 2 , 3 , - 1 ) satisfies equation (2).

Substituting $$\begin{gathered} x=2 \\ y=3 \\ z=-1 \end{gathered}$$ in the equation

$2x-6y+z=3$,

$$\begin{gathered} 2\left(2\right)-6\left(3\right)+(-1)=3 \\ 4-18-1=3 \\ 4-19=3 \\ -15\ne 3 \end{gathered}$$

Part a. Step 3. Finding whether ( 2 , 3 , - 1 ) is a solution of equation (3).

Substituting $$\begin{gathered} x=2 \\ y=3 \\ z=-1 \end{gathered}$$ in the equation

$2x+3y-2z=3$,

$$\begin{gathered} 2\left(2\right)+3\left(3\right)-2(-1)=3 \\ 4+9+2=3 \\ 15\ne 3 \end{gathered}$$

The ordered triple $(2,3,-1)$ is not a solution to the system of linear equations.

Part b. Step 1. Finding whether ( 3 , 1 , 3 ) is a solution of equation (1).

Substituting $$\begin{gathered} x=3 \\ y=1 \\ z=3 \end{gathered}$$ in the equation

$3x-4y-3z=2$,

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

Part b. Step 2. Finding whether ( 3 , 1 , 3 ) is a solution to equation (2).

Substituting $$\begin{gathered} x=3 \\ y=1 \\ z=3 \end{gathered}$$ in the equation

$2x-6y+z=3$,

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

Part b. Step 3. Finding whether ( 3 , 1 , 3 ) is a solution for equation (3).

Substituting $$\begin{gathered} x=3 \\ y=1 \\ z=3 \end{gathered}$$ in the equation

$2x+3y-2z=3$,

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

The ordered triple $(3,1,3)$ is not a solution for the system of equations