To solve the system of equations using the elimination method, we begin with the following equations:
[tex]\[
\begin{array}{l}
2m - 4n = 4 \quad \text{(Equation 1)} \\
m - n = 2 \quad \text{(Equation 2)}
\end{array}
\][/tex]
We aim to eliminate one of the variables by manipulating and combining these equations. Let's start by isolating one variable. Since the coefficient of [tex]\( m \)[/tex] in Equation 2 is smaller than in Equation 1, it may be easier to work with the second equation. Multiply Equation 2 by 2 so that the coefficients of [tex]\( m \)[/tex] in both equations match:
[tex]\[
2(m - n) = 2 \cdot 2
\][/tex]
[tex]\[
2m - 2n = 4 \quad \text{(Equation 3)}
\][/tex]
Now we will subtract Equation 3 from Equation 1:
[tex]\[
(2m - 4n) - (2m - 2n) = 4 - 4
\][/tex]
[tex]\[
2m - 4n - 2m + 2n = 0
\][/tex]
[tex]\[
-2n = 0
\][/tex]
This simplifies to:
[tex]\[
2n = 0
\][/tex]
[tex]\[
n = 0
\][/tex]
With [tex]\( n \)[/tex] identified as 0, we can substitute [tex]\( n \)[/tex] back into either original equation to solve for [tex]\( m \)[/tex]. Using Equation 2:
[tex]\[
m - n = 2
\][/tex]
[tex]\[
m - 0 = 2
\][/tex]
[tex]\[
m = 2
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(m, n) = (2, 0)
\][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[
(2,0)
\][/tex]
