Answer :
Let's analyze the system of equations:
[tex]\[ \begin{array}{l} x + 2y = 6 \quad \text{(Equation 1)} \\ 2x + 4y = 12 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 1: Simplify the System
Notice that Equation 2 is essentially a multiple of Equation 1. If we divide Equation 2 by 2, we get:
[tex]\[ \frac{2x + 4y}{2} = \frac{12}{2} \][/tex]
This simplifies to:
[tex]\[ x + 2y = 6 \][/tex]
which is identical to Equation 1.
### Step 2: Interpret the System
Since both equations are actually the same when simplified, they represent the same line. This means that every point on the line \( x + 2y = 6 \) is also a point on the line \( 2x + 4y = 12 \).
### Step 3: Conclusion
Because the equations describe the same line, there is not just one single solution or no solutions at all. Instead, there are infinitely many solutions, represented by all the points that lie on the line \( x + 2y = 6 \).
### Graphical Interpretation
If we graph these equations, both would appear as the same line on the coordinate plane. Thus, every point \((x, y)\) that satisfies \( x + 2y = 6 \) also satisfies \( 2x + 4y = 12 \).
### Solution to the System
Given that both equations define the same line, the system has infinitely many solutions. Therefore, the correct answer is:
There are infinitely many solutions.
[tex]\[ \begin{array}{l} x + 2y = 6 \quad \text{(Equation 1)} \\ 2x + 4y = 12 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 1: Simplify the System
Notice that Equation 2 is essentially a multiple of Equation 1. If we divide Equation 2 by 2, we get:
[tex]\[ \frac{2x + 4y}{2} = \frac{12}{2} \][/tex]
This simplifies to:
[tex]\[ x + 2y = 6 \][/tex]
which is identical to Equation 1.
### Step 2: Interpret the System
Since both equations are actually the same when simplified, they represent the same line. This means that every point on the line \( x + 2y = 6 \) is also a point on the line \( 2x + 4y = 12 \).
### Step 3: Conclusion
Because the equations describe the same line, there is not just one single solution or no solutions at all. Instead, there are infinitely many solutions, represented by all the points that lie on the line \( x + 2y = 6 \).
### Graphical Interpretation
If we graph these equations, both would appear as the same line on the coordinate plane. Thus, every point \((x, y)\) that satisfies \( x + 2y = 6 \) also satisfies \( 2x + 4y = 12 \).
### Solution to the System
Given that both equations define the same line, the system has infinitely many solutions. Therefore, the correct answer is:
There are infinitely many solutions.
