We are given the following system of linear equations:

\frac{x}{11} - \frac{y}{8} = -2 \tag{1} \frac{x}{2} - \frac{y}{4} = 3 \tag{2}

We can solve this system using substitution or elimination. Let's proceed step by step using the elimination method to simplify the equations.

Step 1: Eliminate the fractions

To make the equations easier to handle, we multiply each equation by the least common multiple (LCM) of the denominators.

• For equation (1), multiply by 88 (LCM of 11 and 8): $88 \times \left( \frac{x}{11} - \frac{y}{8} \right) = 88 \times -2$ 8x - 11y = -176 \tag{3}

• For equation (2), multiply by 4 (LCM of 2 and 4): $4 \times \left( \frac{x}{2} - \frac{y}{4} \right) = 4 \times 3$ 2x - y = 12 \tag{4}

Step 2: Use elimination

We now have the following system:

8x - 11y = -176 \tag{3} 2x - y = 12 \tag{4}

We will eliminate one variable by multiplying equation (4) by 11 to align the coefficients of $y$:

$11 \times (2x - y) = 11 \times 12$ 22x - 11y = 132 \tag{5}

Step 3: Subtract the equations

Now subtract equation (3) from equation (5):

$(22x - 11y) - (8x - 11y) = 132 - (-176)$ $22x - 8x = 132 + 176$ $14x = 308$ $x = \frac{308}{14} = 22$

Step 4: Substitute $x = 22$ into equation (4)

Now that we know $x = 22$, substitute it into equation (4):

$2(22) - y = 12$ $44 - y = 12$ $y = 44 - 12 = 32$

Final Solution:

The solution to the system of equations is: $x = 22, \quad y = 32$

Would you like more details or have any questions? Here are some related problems you can try:

1. Solve the system: $\frac{x}{5} - \frac{y}{3} = 7$ and $\frac{x}{2} + \frac{y}{4} = 1$.
2. Solve the system: $3x - 4y = 9$ and $5x + 2y = 7$.
3. Solve the system using substitution: $x + 2y = 10$ and $3x - y = 5$.
4. Solve the system: $\frac{2x}{3} + \frac{y}{6} = 5$ and $\frac{x}{4} - \frac{y}{3} = 2$.
5. Find the solution of $x - 4y = -1$ and $2x + 3y = 12$.

Tip: When dealing with fractions in systems of equations, multiplying through by the LCM of the denominators can help simplify the process.