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What is the solution to the system of equations?

[tex]\[
\left\{\begin{array}{l}
2x - y + z = -8 \\
x + y + z = -4 \\
3x - y - z = -4
\end{array}\right.
\][/tex]

A. [tex]\((-2, 1, -3)\)[/tex]
B. [tex]\((2, 3, -9)\)[/tex]
C. [tex]\((-2, -5, -9)\)[/tex]
D. [tex]\((2, 21, 9)\)[/tex]



Answer :

GinnyAnswer
To solve the system of linear equations:

[tex]\[ \left\{ \begin{array}{l} 2x - y + z = -8 \\ x + y + z = -4 \\ 3x - y - z = -4 \end{array} \right. \][/tex]

We will follow a systematic approach to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].

### Step 1: Write down the system of equations
[tex]\[ \left\{ \begin{array}{l} 2x - y + z = -8 \quad \text{(1)} \\ x + y + z = -4 \quad \text{(2)} \\ 3x - y - z = -4 \quad \text{(3)} \end{array} \right. \][/tex]

### Step 2: Add Equation (1) and Equation (3) to eliminate [tex]\( y \)[/tex]
[tex]\[ 2x - y + z + 3x - y - z = -8 - 4 \][/tex]
[tex]\[ 5x - 2y = -12 \quad \text{(4)} \][/tex]

### Step 3: Add Equation (1) and Equation (2) to eliminate [tex]\( z \)[/tex]
[tex]\[ 2x - y + z + x + y + z = -8 - 4 \][/tex]
[tex]\[ 3x + 2z = -12 \quad \text{(5)} \][/tex]

### Step 4: Add Equation (2) and Equation (3) to eliminate [tex]\( y \)[/tex]
[tex]\[ x + y + z + 3x - y - z = -4 - 4 \][/tex]
[tex]\[ 4x = -8 \][/tex]
[tex]\[ x = -2 \quad \text{(6)} \][/tex]

### Step 5: Substitute [tex]\( x = -2 \)[/tex] back into Equation (4) and Equation (5)
#### Substitute [tex]\( x = -2 \)[/tex] in Equation (4):
[tex]\[ 5(-2) - 2y = -12 \][/tex]
[tex]\[ -10 - 2y = -12 \][/tex]
[tex]\[ -2y = -2 \][/tex]
[tex]\[ y = 1 \quad \text{(7)} \][/tex]

#### Substitute [tex]\( x = -2 \)[/tex] in Equation (5):
[tex]\[ 3(-2) + 2z = -12 \][/tex]
[tex]\[ -6 + 2z = -12 \][/tex]
[tex]\[ 2z = -6 \][/tex]
[tex]\[ z = -3 \quad \text{(8)} \][/tex]

### Step 6: Summarize the solution
The solution to the system of equations is:
[tex]\[ x = -2, \quad y = 1, \quad z = -3 \][/tex]

Thus, the correct answer is:
[tex]\[ (-2, 1, -3) \][/tex]

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