Let's analyze and solve the given system of equations:
[tex]\[
\left\{\begin{array}{c}
2y + 4x = -8 \\
-y - 2x = 4
\end{array}\right.
\][/tex]
We will solve this system using the substitution or elimination method step-by-step. Let's start with the elimination method.
### Step 1: Align the system of equations
Let's write the system of equations again for clarity:
1. [tex]\(2y + 4x = -8\)[/tex]
2. [tex]\(-y - 2x = 4\)[/tex]
### Step 2: Eliminate one variable
We can eliminate one variable by adding or subtracting the equations. To do this, we'll first multiply the second equation by 2 to make the coefficients of [tex]\(y\)[/tex] equal in magnitude but opposite in sign:
[tex]\[2(-y - 2x) = 2(4)\][/tex]
This results in:
[tex]\[-2y - 4x = 8\][/tex]
Now we have the system:
1. [tex]\(2y + 4x = -8\)[/tex]
2. [tex]\(-2y - 4x = 8\)[/tex]
### Step 3: Add the equations
Next, we add the two equations together to eliminate [tex]\(y\)[/tex]:
[tex]\[
(2y + 4x) + (-2y - 4x) = -8 + 8
\][/tex]
This simplifies to:
[tex]\[
0 = 0
\][/tex]
### Step 4: Interpret the result
The equation [tex]\(0 = 0\)[/tex] is always true, which indicates that the system of equations is dependent, and every point on one line is also a point on the other line.
This means that the two equations represent the same line, and therefore, there are infinite solutions to this system.
### Conclusion
The given system of equations does not have a single unique solution or no solution. Because it is consistent and dependent, the solution is:
[tex]\[
\text{Infinite Solutions}
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{Infinite Solutions}
\][/tex]
