To solve the system of equations using the elimination method, we have the following equations:
- \( 2x + 4y = 5 \) (Equation 1)
- \( -2x - y = 4 \) (Equation 2)
To eliminate \(x\), we can add Equation 1 and Equation 2 together. First, we can express Equation 2 by multiplying it by 1 (to keep it unchanged):
\[ -2x - y = 4 \]
Now let's add the two equations:
\[ (2x + 4y) + (-2x - y) = 5 + 4 \]
When we combine like terms, the \(2x\) and \(-2x\) cancel out:
\[ (2x - 2x) + (4y - y) = 9 \]
This simplifies to:
\[ 3y = 9 \]
Now, solve for \(y\):
\[ y = \frac{9}{3} = 3 \]
Next, we will substitute \(y = 3\) back into either of the original equations to solve for \(x\). We'll use Equation 1:
\[ 2x + 4(3) = 5 \]
This simplifies to:
\[ 2x + 12 = 5 \]
Now, isolate \(x\):
\[ 2x = 5 - 12 \]
\[ 2x = -7 \]
\[ x = \frac{-7}{2} \]
Thus, the solution to the system of equations is:
\[ \left(-\frac{7}{2}, 3\right) \]
So, the correct answer among the options provided is:
The solution is \((-7/2, 3)\).