We are given the following system of equations:

$1. \quad x + \frac{1}{3}y - 2z = -1$ $2. \quad \frac{1}{3}x + y + \frac{1}{2}z = 0$ $3. \quad \frac{1}{2}x + \frac{1}{3}y - \frac{1}{2}z = -1$

We will solve this system using substitution or elimination. First, let's convert these equations into a more standard form.

Step 1: Eliminate fractions by multiplying each equation by an appropriate factor.

• Multiply equation 1 by 3: $3x + y - 6z = -3 \quad \text{(Equation 1')}$
• Multiply equation 2 by 6: $2x + 6y + 3z = 0 \quad \text{(Equation 2')}$
• Multiply equation 3 by 6: $3x + 2y - 3z = -6 \quad \text{(Equation 3')}$

The new system is:

$1'. \quad 3x + y - 6z = -3$ $2'. \quad 2x + 6y + 3z = 0$ $3'. \quad 3x + 2y - 3z = -6$

Step 2: Solve the system using elimination.

Let's eliminate $x$ by subtracting equations.

Subtract Equation 2' from Equation 1':

$(3x + y - 6z) - (2x + 6y + 3z) = -3 - 0$ $x - 5y - 9z = -3 \quad \text{(Equation 4)}$

Subtract Equation 3' from Equation 1':

$(3x + y - 6z) - (3x + 2y - 3z) = -3 - (-6)$ $-y - 3z = 3 \quad \text{(Equation 5)}$ $y + 3z = -3 \quad \text{(Rearranged Equation 5)}$

Step 3: Substitute and solve.

We now have two new equations: $4. \quad x - 5y - 9z = -3$ $5. \quad y + 3z = -3$

Substitute $y = -3 - 3z$ (from Equation 5) into Equation 4: $x - 5(-3 - 3z) - 9z = -3$ $x + 15 + 15z - 9z = -3$ $x + 6z = -18$ $x = -18 - 6z \quad \text{(Equation 6)}$

Step 4: Find the solution.

Now, we have $x = -18 - 6z$ and $y = -3 - 3z$. These are parametric solutions in terms of $z$, meaning there are infinitely many solutions.

Conclusion:

Since we have a parametric solution for $x$, $y$, and $z$ in terms of $z$, the system has infinitely many solutions.

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Would you like further details or have any questions?

Here are 5 related problems you might find interesting:

1. Solve a system of equations with two variables.
2. Solve a system of three linear equations using matrices.
3. Solve a system of nonlinear equations.
4. Solve a system of equations using Gaussian elimination.
5. Determine whether a system of equations is inconsistent.

Tip: When solving systems of equations, always check for special cases like "no solution" or "infinitely many solutions" based on the consistency of the equations.