To solve the system of equations using the elimination method, we can start with the given equations:
- \(-x + 6y = 13\) (Equation 1)
- \(4x - 10y = -10\) (Equation 2)
First, let's manipulate the equations to eliminate one of the variables. We can multiply Equation 1 by 4 in order to align the coefficients of \(x\) for elimination:
\[ 4(-x + 6y) = 4(13) \]
This gives us:
\[ -4x + 24y = 52 \quad \text{(Equation 3)} \]
Now we have:
- \(-4x + 24y = 52\) (Equation 3)
- \(4x - 10y = -10\) (Equation 2)
Next, we can add Equation 2 and Equation 3 together to eliminate \(x\):
\[ (-4x + 24y) + (4x - 10y) = 52 + (-10) \]
This simplifies to:
\[ (24y - 10y) = 42 \]
So we have:
\[ 14y = 42 \]
Now, divide both sides by 14:
\[ y = \frac{42}{14} = 3 \]
Now that we have the value of \(y\), we can substitute it back into one of the original equations to find \(x\). Let's use Equation 1:
\[ -x + 6(3) = 13 \]
This simplifies to:
\[ -x + 18 = 13 \]
Now, isolate \(x\):
\[ -x = 13 - 18 \] \[ -x = -5 \]
Thus:
\[ x = 5 \]
So the solution to the system of equations is:
\[ (x, y) = (5, 3) \]
In coordinate form, the solution is:
(5, 3).