To solve the system of equations:
- \(-x + 6y = 13\) (Equation 1)
- \(4x - 10y = -10\) (Equation 2)
We can use the substitution or elimination method. We'll use the substitution method here.
First, solve Equation 1 for \(x\):
\[ -x + 6y = 13 \implies -x = 13 - 6y \implies x = 6y - 13 \]
Now we can substitute \(x\) in Equation 2:
\[ 4(6y - 13) - 10y = -10 \]
Distributing the \(4\):
\[ 24y - 52 - 10y = -10 \]
Combine like terms:
\[ (24y - 10y) - 52 = -10 \implies 14y - 52 = -10 \]
Now, add \(52\) to both sides:
\[ 14y = 42 \]
Now, divide both sides by \(14\):
\[ y = 3 \]
Now we can substitute \(y = 3\) back into the equation we found for \(x\):
\[ x = 6(3) - 13 = 18 - 13 = 5 \]
Thus, the solution to the system of equations is:
\[ \boxed{(5, 3)} \]