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Which system of equations is represented by the matrix below?

[tex]\[
\left[\begin{array}{ll|c}
2 & 4 & 12 \\
2 & 0 & 4
\end{array}\right]
\][/tex]

A. [tex]\(2x + 4y = 12\)[/tex] \\
[tex]\(2x = 4\)[/tex]

B. [tex]\(2x + 4y = 12\)[/tex]

C. [tex]\(2x - 4y = 12\)[/tex] \\
[tex]\(2y = 4\)[/tex]

D. [tex]\(2x - 4y = 12\)[/tex] \\
[tex]\(y = 4\)[/tex]

Sagot :

GinnyAnswer
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.
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