To find the solution to the system of two linear equations
- \( x - 5y = 7 \)
- \( 3x - 2y = -4 \)
we can solve the equations using the substitution or elimination method. Here, I will use the substitution method.
Step 1: Solve the first equation for \( x \).
From the first equation:
\[ x - 5y = 7 \]
We can rearrange it to solve for \( x \):
\[ x = 5y + 7 \]
Step 2: Substitute \( x \) in the second equation.
Now substitute \( x \) in the second equation \( 3x - 2y = -4 \):
\[ 3(5y + 7) - 2y = -4 \]
Step 3: Simplify and solve for \( y \).
Distributing the 3:
\[ 15y + 21 - 2y = -4 \]
Combine like terms:
\[ 13y + 21 = -4 \]
Next, subtract 21 from both sides:
\[ 13y = -4 - 21 \]
\[ 13y = -25 \]
Now, divide by 13:
\[ y = -\frac{25}{13} \]
Step 4: Substitute \( y \) back to find \( x \).
Now plug the value of \( y \) back into the expression for \( x \):
\[ x = 5\left(-\frac{25}{13}\right) + 7 \]
\[ x = -\frac{125}{13} + 7 \]
Convert 7 to have a common denominator:
\[ x = -\frac{125}{13} + \frac{91}{13} \]
Combine the fractions:
\[ x = \frac{-125 + 91}{13} = \frac{-34}{13} \]
Conclusion:
The solution to the system of equations is:
\[ \left( x, y \right) = \left( -\frac{34}{13}, -\frac{25}{13} \right) \]
Thus, the final answer is:
\[ \boxed{\left(-\frac{34}{13}, -\frac{25}{13}\right)} \]