To solve the system of equations:
- \(-x + 6y = 13\)
- \(4x - 10y = -10\)
We can use the substitution or elimination method. Here, I will use the substitution method for clarity.
First, let's solve the first equation for \(x\):
\[ -x + 6y = 13 \implies -x = 13 - 6y \implies x = 6y - 13 \]
Now, we will substitute \(x\) into the second equation:
\[ 4(6y - 13) - 10y = -10 \]
Expanding the equation:
\[ 24y - 52 - 10y = -10 \]
Combining like terms:
\[ 24y - 10y - 52 = -10 \implies 14y - 52 = -10 \]
Now, add 52 to both sides:
\[ 14y = -10 + 52 \implies 14y = 42 \]
Dividing both sides by 14:
\[ y = \frac{42}{14} = 3 \]
Now that we have \(y\), we substitute it back into our expression for \(x\):
\[ x = 6y - 13 = 6(3) - 13 = 18 - 13 = 5 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (5, 3) \]
To confirm, we can substitute \(x\) and \(y\) back into the original equations:
- \(-5 + 6(3) = -5 + 18 = 13\) (True)
- \(4(5) - 10(3) = 20 - 30 = -10\) (True)
Both equations are satisfied, therefore the solution is correct:
\[ \boxed{(5, 3)} \]