Let’s analyze the given matrix step-by-step to identify the correct system of equations it represents.
The given matrix is:
[tex]\[
\left[\begin{array}{ll|c}
2 & 4 & 12 \\
2 & 0 & 4
\end{array}\right]
\][/tex]
### Step 1: Interpreting the Matrix
This matrix indicates that we have a system of two linear equations. Each row represents one equation. The format of the matrix is:
[tex]\[
\left[\begin{array}{ll|c}
a_{11} & a_{12} & b_1 \\
a_{21} & a_{22} & b_2
\end{array}\right]
\][/tex]
where [tex]\(a_{ij}\)[/tex] represents the coefficients of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and [tex]\(b_i\)[/tex] represents the constants on the right side of the equations.
### Step 2: Writing the Equations
From the first row:
[tex]\[2x + 4y = 12\][/tex]
From the second row:
[tex]\[2x + 0y = 4\][/tex]
### Step 3: Simplifying the Equations
The first equation is already in a simple, interpretable form:
[tex]\[2x + 4y = 12\][/tex]
The second equation simplifies to:
[tex]\[2x = 4\][/tex]
Dividing both sides by 2:
[tex]\[x = 2\][/tex]
### Step 4: Comparing with Given Options
Now, let's compare these findings with the provided options:
Option A:
[tex]\[2x + 4y = 12\][/tex]
[tex]\[2x = 4\][/tex]
This matches exactly with our found equations.
Option B:
[tex]\[2x + 4y = 12\][/tex]
This does not account for the second equation [tex]\(2x = 4\)[/tex], so it is incomplete.
Option C:
[tex]\[2x - 4y = 12\][/tex]
[tex]\[2y = 4\][/tex]
This system does not match either of our equations as the signs and structure differ significantly.
Option D:
[tex]\[2x - 4y = 12\][/tex]
[tex]\[y = 4\][/tex]
This also does not match our equations.
### Conclusion
The system of equations represented by the given matrix is:
[tex]\[
\left\{
\begin{align*}
2x + 4y &= 12 \\
2x &= 4
\end{align*}
\right.
\][/tex]
Hence, the correct answer is Option A.
