Which type of lines are represented by the pair of linear equations `3x + 8y =13` and `21x + 56y = 5?`

Options

A. Coincident
B. Intersecting at exactly one point
C. Parallel
D. Intersecting at two points

Answer

To determine the type of lines represented by the pair of linear equations \(3x + 8y = 13\) and \(21x + 56y = 5\), we can follow these steps:

### Step 1: Identify the coefficients
From the equations, we can identify the coefficients:
- For the first equation \(3x + 8y = 13\):
- \(a_1 = 3\)
- \(b_1 = 8\)
- \(c_1 = 13\)

- For the second equation \(21x + 56y = 5\):
- \(a_2 = 21\)
- \(b_2 = 56\)
- \(c_2 = 5\)

### Step 2: Calculate the ratios
Now, we calculate the ratios of the coefficients:
1. \( \frac{a_1}{a_2} = \frac{3}{21} = \frac{1}{7} \)
2. \( \frac{b_1}{b_2} = \frac{8}{56} = \frac{1}{7} \)
3. \( \frac{c_1}{c_2} = \frac{13}{5} \)

### Step 3: Compare the ratios
Now we compare the ratios:
- \( \frac{a_1}{a_2} = \frac{1}{7} \)
- \( \frac{b_1}{b_2} = \frac{1}{7} \)
- \( \frac{c_1}{c_2} = \frac{13}{5} \)

### Step 4: Analyze the results
From our calculations:
- \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) (both equal to \(\frac{1}{7}\))
- \( \frac{c_1}{c_2} \) is not equal to the other two ratios.

### Conclusion
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal to these, we conclude that the lines represented by the equations are **parallel lines**.

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