To find the solution to the system of equations given by the lines \( x - 5y = 7 \) and \( 3x - 2y = -4 \), we need to solve these equations simultaneously.
Step 1: Solve for \( y \) in terms of \( x \) using the first equation.
Starting with the equation:
\[ x - 5y = 7 \]
Rearranging to isolate \( y \):
\[ -5y = 7 - x \]
\[ y = \frac{x - 7}{5} \]
Step 2: Substitute \( y \) in the second equation.
Now we will substitute this expression for \( y \) into the second equation \( 3x - 2y = -4 \):
\[ 3x - 2\left(\frac{x - 7}{5}\right) = -4 \]
Step 3: Clear the fraction by multiplying the entire equation by 5.
\[ 5(3x) - 2(x - 7) = -4 \cdot 5 \]
\[ 15x - 2(x - 7) = -20 \]
Expanding the equation:
\[ 15x - 2x + 14 = -20 \]
Combining like terms:
\[ 13x + 14 = -20 \]
Step 4: Solve for \( x \).
Subtracting 14 from both sides gives:
\[ 13x = -20 - 14 \]
\[ 13x = -34 \]
Dividing by 13:
\[ x = -\frac{34}{13} \]
Step 5: Substitute \( x \) back to find \( y \).
Now substitute \( x = -\frac{34}{13} \) back into the equation for \( y \):
\[ y = \frac{-\frac{34}{13} - 7}{5} \]
Converting 7 to a fraction with a common denominator of 13:
\[ y = \frac{-\frac{34}{13} - \frac{91}{13}}{5} \]
\[ y = \frac{-\frac{125}{13}}{5} \]
\[ y = -\frac{125}{65} = -\frac{25}{13} \]
Step 6: Write the solution as an ordered pair.
Thus, the intersection point (solution to the system of equations) is:
\[ \left(-\frac{34}{13}, -\frac{25}{13}\right) \]
Final answer: The intersection point is \(\left(-\frac{34}{13}, -\frac{25}{13}\right)\).