The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 is :

Options

A. `(10)/(3)`
B. `(3)/(10)`
C. `(1)/(2)`
D. 2

Answer

To find the distance between the lines given by the equations \(3x + 4y = 9\) and \(6x + 8y = 15\), we can follow these steps:

### Step 1: Write down the equations of the lines.
The first line is:
\[ 3x + 4y = 9 \]
The second line is:
\[ 6x + 8y = 15 \]

### Step 2: Simplify the second line.
We can simplify the second line by dividing the entire equation by 2:
\[ \frac{6x}{2} + \frac{8y}{2} = \frac{15}{2} \]
This simplifies to:
\[ 3x + 4y = \frac{15}{2} \]

### Step 3: Identify the coefficients and constants.
Now we have two lines:
1. \(3x + 4y = 9\) (here, \(c_1 = 9\))
2. \(3x + 4y = \frac{15}{2}\) (here, \(c_2 = \frac{15}{2}\))

The coefficients of \(x\) and \(y\) are:
- \(a = 3\)
- \(b = 4\)

### Step 4: Use the distance formula.
The formula to find the distance \(d\) between two parallel lines \(Ax + By = C_1\) and \(Ax + By = C_2\) is given by:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ d = \frac{|9 - \frac{15}{2}|}{\sqrt{3^2 + 4^2}} \]

### Step 5: Calculate the numerator.
First, we need to find \(9 - \frac{15}{2}\):
\[ 9 = \frac{18}{2} \quad \text{(to have a common denominator)} \]
Thus:
\[ 9 - \frac{15}{2} = \frac{18}{2} - \frac{15}{2} = \frac{3}{2} \]
So, the absolute value is:
\[ |9 - \frac{15}{2}| = \frac{3}{2} \]

### Step 6: Calculate the denominator.
Now, we calculate \(\sqrt{3^2 + 4^2}\):
\[ 3^2 = 9, \quad 4^2 = 16 \quad \Rightarrow \quad 3^2 + 4^2 = 9 + 16 = 25 \]
Thus:
\[ \sqrt{25} = 5 \]

### Step 7: Substitute back into the distance formula.
Now we can substitute back into the formula for \(d\):
\[ d = \frac{\frac{3}{2}}{5} = \frac{3}{2 \times 5} = \frac{3}{10} \]

### Final Answer:
The distance between the two lines is:
\[ \boxed{\frac{3}{10}} \]