To solve the system of equations using the elimination method, we start with the following equations:
-
\(-3x - 3y = 3\)
-
\(3x + 6y = 18\)
Next, we can manipulate the equations to eliminate one variable. A convenient way to eliminate \(x\) is to add the two equations together, but first, let's manipulate the first equation so we can see if we can easily eliminate \(x\):
- \(-3x - 3y = 3\) can be simplified by dividing every term by -3:
\[ x + y = -1 \quad \text{(Equation 1')} \]
Now, we can rewrite the second equation:
- \(3x + 6y = 18\) can be simplified by dividing every term by 3:
\[ x + 2y = 6 \quad \text{(Equation 2')} \]
Now, we have a new system of equations:
1') \(x + y = -1\)
2') \(x + 2y = 6\)
Next, we'll eliminate \(x\) by subtracting equation 1' from equation 2':
\[ (x + 2y) - (x + y) = 6 - (-1) \]
This simplifies to:
\[ 2y - y = 6 + 1 \] \[ y = 7 \]
Now that we have \(y\), we can substitute it back into equation 1' to find \(x\):
\[ x + 7 = -1 \] \[ x = -1 - 7 \] \[ x = -8 \]
Thus, the solution to the system of equations is:
\[ \boxed{(-8, 7)} \]