To solve the system of equations using the elimination method, we start with the two equations:
- \(-x + 6y = 13\)
- \(4x - 10y = -10\)
First, we can clear one of the variables by manipulating the equations. Let's eliminate \(x\) by making the coefficients of \(x\) the same in both equations.
We can do this by multiplying the first equation by 4:
\[ 4(-x + 6y) = 4(13) \]
This gives us:
\[ -4x + 24y = 52 \quad (3) \]
Now we have the rewritten system of equations:
\[ -4x + 24y = 52 \quad (3) \] \[ 4x - 10y = -10 \quad (2) \]
Next, we can add equations (2) and (3) to eliminate \(x\):
\[ (-4x + 24y) + (4x - 10y) = 52 - 10 \]
This simplifies to:
\[ (24y - 10y) = 42 \]
So:
\[ 14y = 42 \]
Now, divide both sides by 14:
\[ y = 3 \]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). Let’s use the first equation:
\[ -x + 6(3) = 13 \]
This simplifies to:
\[ -x + 18 = 13 \]
Now, isolate \(x\):
\[ -x = 13 - 18 \] \[ -x = -5 \]
So:
\[ x = 5 \]
Hence, the solution to the system of equations is:
\[ (x, y) = (5, 3) \]
Final answer:
\[ (5, 3) \]