Which type of lines are represented by the pair of linear equations
4x + 3y - 1 = 5 and 12x + 9y = 15.

Options

A. Coincident
B. Intersecting
C. Parallel
D. both (a) and ( c)

Answer

To determine the type of lines represented by the pair of linear equations \(4x + 3y - 1 = 5\) and \(12x + 9y = 15\), we will follow these steps:

### Step 1: Rewrite the equations in standard form
First, we will rewrite the first equation \(4x + 3y - 1 = 5\) in standard form.

\[ 4x + 3y - 1 = 5 \implies 4x + 3y = 6 \quad \text{(Equation 1)} \]

The second equation is already in standard form:

\[ 12x + 9y = 15 \quad \text{(Equation 2)} \]

### Step 2: Identify coefficients
From the equations, we identify the coefficients:
- For Equation 1: \(a_1 = 4\), \(b_1 = 3\), \(c_1 = -6\) (after moving 6 to the right side)
- For Equation 2: \(a_2 = 12\), \(b_2 = 9\), \(c_2 = -15\) (after moving 15 to the right side)

### Step 3: Calculate the ratios of coefficients
Now we will calculate the ratios of the coefficients:

\[ \frac{a_1}{a_2} = \frac{4}{12} = \frac{1}{3} \]

\[ \frac{b_1}{b_2} = \frac{3}{9} = \frac{1}{3} \]

\[ \frac{c_1}{c_2} = \frac{6}{15} = \frac{2}{5} \]

### Step 4: Analyze the ratios
Now we compare the ratios:
- \(\frac{a_1}{a_2} = \frac{1}{3}\)
- \(\frac{b_1}{b_2} = \frac{1}{3}\)
- \(\frac{c_1}{c_2} = \frac{2}{5}\)

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

### Conclusion
The type of lines represented by the given pair of linear equations is **parallel lines**.

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