To solve the system of equations:
[tex]\[
\begin{cases}
\frac{2}{3x} - \frac{1}{y} = -\frac{1}{6} & \quad \text{(1)} \\
\frac{5}{x} - \frac{6}{y} = -\frac{3}{4} & \quad \text{(2)}
\end{cases}
\][/tex]
we need to find values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Rewrite Equation (1):
[tex]\[
\frac{2}{3x} - \frac{1}{y} = -\frac{1}{6}
\][/tex]
To make the equation more manageable, let's eliminate the fractions by finding a common denominator. However, let's just keep it as is for solving.
2. Rewrite Equation (2):
[tex]\[
\frac{5}{x} - \frac{6}{y} = -\frac{3}{4}
\][/tex]
3. Determine a common approach:
One strategy is solving each equation for one variable in terms of the other and then substitute back into the other equation. But as we proceed, we realize we need to simplify and combine both equations effectively.
4. Finding the solution:
Upon solving the above system consistently, especially focusing on aligning coefficients and solving for one variable in terms of another, we reach the numerical solution:
[tex]\[
x = 4, \quad y= 3
\][/tex]
Thus, the solution to the given system of equations is:
[tex]\[
x = 4 \quad \text{and} \quad y = 3
\][/tex]
