To solve the system of equations using the elimination method, we will eliminate one of the variables by adding or subtracting the equations.
The system of equations is:
- \(3x - y = 7\) (Equation 1)
- \(-2x + y = 2\) (Equation 2)
We can add the two equations together to eliminate \(y\):
\[ (3x - y) + (-2x + y) = 7 + 2 \]
Now, simplify the left-hand side:
\[ 3x - 2x - y + y = 9 \]
This simplifies to:
\[ x = 9 \]
Now that we have \(x\), we can substitute it back into either equation to find \(y\). We'll use Equation 1:
\[ 3(9) - y = 7 \] \[ 27 - y = 7 \] \[ -y = 7 - 27 \] \[ -y = -20 \] \[ y = 20 \]
Thus, the solution to the system of equations is:
\[ x = 9, \quad y = 20 \]
So, the solution is \((9, 20)\).