To find the solution to the system of linear equations given by:
- \( x - 5y = 7 \)
- \( 3x - 2y = -4 \)
we can solve these equations using either the substitution method, the elimination method, or by graphing them. Here, I'll use the substitution method for clarity.
Step 1: Solve one equation for one variable
Let's solve the first equation for \( x \):
\[ x = 5y + 7 \]
Step 2: Substitute into the second equation
Next, we substitute this expression for \( x \) into the second equation:
\[ 3(5y + 7) - 2y = -4 \]
Step 3: Simplify and solve for \( y \)
Expanding this gives:
\[ 15y + 21 - 2y = -4 \]
Now combine like terms:
\[ 13y + 21 = -4 \]
Subtracting 21 from both sides results in:
\[ 13y = -4 - 21 \] \[ 13y = -25 \]
Now, divide by 13 to solve for \( y \):
\[ y = -\frac{25}{13} \]
Step 4: Substitute back to find \( x \)
Now that we have \( y \), we can substitute back to find \( x \):
\[ x = 5\left(-\frac{25}{13}\right) + 7 \] \[ x = -\frac{125}{13} + \frac{91}{13} = -\frac{125 - 91}{13} = -\frac{34}{13} \]
Conclusion
So the solution to the system of equations is:
\[ x = -\frac{34}{13}, \quad y = -\frac{25}{13} \]
In ordered pair notation, the solution is:
\[ \left(-\frac{34}{13}, -\frac{25}{13}\right) \]