To solve the system of equations using the elimination method, we can start by rewriting the equations:
- \(3x + 6y = 21\) (Equation 1)
- \(-5x + 2y = 13\) (Equation 2)
First, we can simplify Equation 1. Dividing all terms by 3:
\[ x + 2y = 7 \quad \text{(Simplified Equation 1)} \]
Now we have:
- \(x + 2y = 7\)
- \(-5x + 2y = 13\)
Next, let's eliminate \(y\) by subtracting Equation 1 from Equation 2. We can rewrite Equation 1 and then subtract:
Subtract Equation 1 from Equation 2:
\[ (-5x + 2y) - (x + 2y) = 13 - 7 \]
This simplifies to:
\[ -5x - x + 2y - 2y = 6 \]
Combining like terms gives:
\[ -6x = 6 \]
Dividing by -6:
\[ x = -1 \]
Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). We can use the simplified Equation 1:
\[ x + 2y = 7 \]
Substituting \(x = -1\):
\[ -1 + 2y = 7 \]
Adding 1 to both sides:
\[ 2y = 8 \]
Dividing by 2:
\[ y = 4 \]
So, the solution to the system of equations is:
\[ (x, y) = (-1, 4) \]
Final answer:
\[ \boxed{(-1, 4)} \]