How many pairs of x and y satisfy the equations `2x + 4y = 8 and 6x + 12y = 24` ?

Options

A. 0
B. 1
C. Infinite
D. None of these

Explanation

To determine how many pairs of \(x\) and \(y\) satisfy the equations \(2x + 4y = 8\) and \(6x + 12y = 24\), we can analyze the relationships between the coefficients of the equations.

### Step 1: Identify the coefficients
For the first equation \(2x + 4y = 8\):
- \(a_1 = 2\)
- \(b_1 = 4\)
- \(c_1 = 8\)

For the second equation \(6x + 12y = 24\):
- \(a_2 = 6\)
- \(b_2 = 12\)
- \(c_2 = 24\)

### Step 2: Calculate the ratios of the coefficients
Now we will calculate the ratios of the coefficients:
- \(\frac{a_1}{a_2} = \frac{2}{6} = \frac{1}{3}\)
- \(\frac{b_1}{b_2} = \frac{4}{12} = \frac{1}{3}\)
- \(\frac{c_1}{c_2} = \frac{8}{24} = \frac{1}{3}\)

### Step 3: Analyze the ratios
We observe that:
- \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{3}\)

Since all three ratios are equal, we conclude that the two equations represent the same line in the coordinate plane.

### Step 4: Conclusion on the number of solutions
When two linear equations represent the same line, there are infinitely many solutions (pairs of \(x\) and \(y\)) that satisfy both equations.

### Final Answer
Thus, the number of pairs \((x, y)\) that satisfy the equations is **infinite**.

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