To find the solution to the system of equations given by:
- \( x - 5y = 7 \) (Equation 1)
- \( 3x - 2y = -4 \) (Equation 2)
we will solve these equations simultaneously.
Step 1: Solve Equation 1 for \( x \)
From Equation 1:
\[ x = 5y + 7 \]
Step 2: Substitute for \( x \) in Equation 2
Now, substitute \( x \) in Equation 2:
\[ 3(5y + 7) - 2y = -4 \]
Step 3: Simplify and Solve for \( y \)
Expanding gives:
\[ 15y + 21 - 2y = -4 \]
Combine like terms:
\[ 13y + 21 = -4 \]
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, substitute \( y \) back into the expression we found for \( x \):
\[ x = 5\left(-\frac{25}{13}\right) + 7 \]
Calculating \( x \):
\[ x = -\frac{125}{13} + 7 \]
Convert 7 to a fraction with a common denominator:
\[ x = -\frac{125}{13} + \frac{91}{13} \]
Combining gives:
\[ x = \frac{-125 + 91}{13} = \frac{-34}{13} \]
Conclusion
The solution to the system of equations is:
\[ \left( \frac{-34}{13}, -\frac{25}{13} \right) \]
Thus, the intersection point is:
The intersection point is \( \left( -\frac{34}{13}, -\frac{25}{13} \right) \).