To find the solution to the system of equations given by:
- \( x - 5y = 7 \)
- \( 3x - 2y = -4 \)
we can use either the substitution method or the elimination method. Here, I'll use the substitution method.
Step 1: Solve the first equation for \( x \)
From the first equation:
\[ x = 5y + 7 \]
Step 2: Substitute for \( x \) in the second equation
Now, substitute \( x \) from the first equation into the second equation:
\[ 3(5y + 7) - 2y = -4 \]
Step 3: Distribute and simplify
Distributing \( 3 \):
\[ 15y + 21 - 2y = -4 \]
Combine like terms:
\[ (15y - 2y) + 21 = -4 \] \[ 13y + 21 = -4 \]
Step 4: Isolate \( y \)
Subtract \( 21 \) from both sides:
\[ 13y = -4 - 21 \] \[ 13y = -25 \]
Now, divide both sides by \( 13 \):
\[ y = -\frac{25}{13} \]
Step 5: Substitute back to find \( x \)
Substitute \( y \) back into the equation \( x = 5y + 7 \):
\[ x = 5\left(-\frac{25}{13}\right) + 7 \]
Calculating \( 5 \cdot -\frac{25}{13} \):
\[ x = -\frac{125}{13} + 7 \] \[ x = -\frac{125}{13} + \frac{91}{13} \] \[ x = -\frac{125 - 91}{13} \] \[ x = -\frac{34}{13} \]
Final Solution
The solution to the system of equations is:
\[ \left( -\frac{34}{13}, -\frac{25}{13} \right) \]
This means that the point \( \left( -\frac{34}{13}, -\frac{25}{13} \right) \) is the point where the two lines intersect, which is the solution to the system.