What is the distance between the lines 3x + 4y = 9 and 6x + 8y = 18 ?

Options

A. 0
B. 3 units
C. 9 units
D. 18 units

Answer

To find the distance between the lines given by the equations \(3x + 4y = 9\) and \(6x + 8y = 18\), we can follow these steps:

### Step 1: Identify the equations of the lines
We have two equations:
1. \(3x + 4y = 9\) (Equation 1)
2. \(6x + 8y = 18\) (Equation 2)

### Step 2: Rewrite the second equation
To see if the two lines are parallel or the same, we can simplify Equation 2. We can divide the entire equation by 2:
\[ \frac{6x + 8y}{2} = \frac{18}{2} \]
This simplifies to:
\[ 3x + 4y = 9 \]
Let’s call this Equation 3.

### Step 3: Compare the equations
Now we have:
- Equation 1: \(3x + 4y = 9\)
- Equation 3: \(3x + 4y = 9\)

Since both equations are identical, it means that the two lines are actually the same line.

### Step 4: Determine the distance between the lines
When two lines are the same, the distance between them is zero. Thus, we can conclude:
\[ \text{Distance} = 0 \]

### Final Answer
The distance between the lines \(3x + 4y = 9\) and \(6x + 8y = 18\) is \(0\) units.

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