We are given a system of three equations with three variables:

2x - 3y + z = -7 \tag{1} 2x + 3y + z = -5 \tag{2} -4x + 9y - 3z = 13 \tag{3}

Step 1: Subtract Equation (2) from Equation (1)

Subtracting $(2x + 3y + z = -5)$ from $(2x - 3y + z = -7)$:

$(2x - 3y + z) - (2x + 3y + z) = -7 - (-5)$ $2x - 3y + z - 2x - 3y - z = -7 + 5$ $-6y = -2 \quad \Rightarrow \quad y = \frac{1}{3}$

Step 2: Substitute $y = \frac{1}{3}$ into Equation (1) and Equation (2)

Substituting $y = \frac{1}{3}$ into Equation (1):

$2x - 3\left(\frac{1}{3}\right) + z = -7$ 2x - 1 + z = -7 \quad \Rightarrow \quad 2x + z = -6 \tag{4}

Substituting $y = \frac{1}{3}$ into Equation (2):

$2x + 3\left(\frac{1}{3}\right) + z = -5$ 2x + 1 + z = -5 \quad \Rightarrow \quad 2x + z = -6 \tag{5}

Step 3: Combine Equations (4) and (5)

Notice that both Equation (4) and Equation (5) give $2x + z = -6$, so this equation holds true.

Step 4: Solve for $x$ and $z$

Substitute $2x + z = -6$ into Equation (3) to solve for $x$ and $z$.

Substituting $y = \frac{1}{3}$ into Equation (3):

$-4x + 9\left(\frac{1}{3}\right) - 3z = 13$ -4x + 3 - 3z = 13 \quad \Rightarrow \quad -4x - 3z = 10 \tag{6}

Now, use Equation (4) $2x + z = -6$ to express $z$ in terms of $x$:

$z = -6 - 2x$

Substitute this into Equation (6):

$-4x - 3(-6 - 2x) = 10$ $-4x + 18 + 6x = 10$ $2x + 18 = 10 \quad \Rightarrow \quad 2x = -8 \quad \Rightarrow \quad x = -4$

Step 5: Solve for $z$

Now substitute $x = -4$ into $2x + z = -6$:

$2(-4) + z = -6 \quad \Rightarrow \quad -8 + z = -6 \quad \Rightarrow \quad z = 2$

Final Solution:

We have $x = -4$, $y = \frac{1}{3}$, and $z = 2$. Therefore, the solution is:

$\boxed{B \left( -4, \frac{1}{3}, 2 \right)}$

Would you like further details or have any questions?

Related Questions:

1. What are the methods to solve systems of equations?
2. How do you solve a system of equations with matrices?
3. What is the substitution method in solving systems of equations?
4. How does Gaussian elimination work for solving systems of equations?
5. Can you use Cramer's rule to solve this system of equations?

Tip:

When solving a system of equations, always check if subtracting or adding equations simplifies the process.