Solve this system of equations \( \left\{\begin{array}{l}4 x+5 y=6 \\ -7 x-5 y= \\ -7 y\end{array}\right. \)

To solve the system of equations: 1. Rewrite the second equation by simplifying it: \(-7x - 5y = -7y\) can be rearranged as \(-7x = -7y + 5y\), which simplifies to \(-7x = -2y\) or \(7x = 2y\). This can be expressed as \(y = \frac{7}{2}x\). 2. Now substitute \(y\) in the first equation: \(4x + 5\left(\frac{7}{2}x\right) = 6\). This simplifies to \(4x + \frac{35}{2}x = 6\). Convert \(4x\) to have a denominator of 2: \(\frac{8}{2}x + \frac{35}{2}x = 6\). Combine the terms: \(\frac{43}{2}x = 6\). 3. Solve for \(x\): \[ x = 6 \cdot \frac{2}{43} = \frac{12}{43}. \] 4. Now use this value to find \(y\): \[ y = \frac{7}{2} \cdot \frac{12}{43} = \frac{84}{86} = \frac{42}{43}. \] So, the solution to the system of equations is: \[ x = \frac{12}{43}, \quad y = \frac{42}{43}. \]