What is the equation of the line midway between the lines 3x - 4y + 12 = 0 and 3x - 4y = 6 ?

Options

A. 3x - 4y -9 = 0
B. 3x - 4y + 9 =0
C. 3x - 4y - 3 = 0
D. 3x - 4y + 3 = 0

Answer

To find the equation of the line that is midway between the lines \(3x - 4y + 12 = 0\) and \(3x - 4y = 6\), we can follow these steps:

### Step 1: Identify the equations of the lines
The two lines given are:
1. Line 1: \(3x - 4y + 12 = 0\)
2. Line 2: \(3x - 4y = 6\)

### Step 2: Rewrite the equations in slope-intercept form
We can rewrite both equations in the form \(y = mx + b\) to easily identify the y-intercepts.

**For Line 1:**
\[ 3x - 4y + 12 = 0 \implies -4y = -3x - 12 \implies y = \frac{3}{4}x + 3 \]
The y-intercept (b) of Line 1 is \(3\).

**For Line 2:**
\[ 3x - 4y = 6 \implies -4y = -3x + 6 \implies y = \frac{3}{4}x - \frac{3}{2} \]
The y-intercept (b) of Line 2 is \(-\frac{3}{2}\).

### Step 3: Calculate the midpoint of the y-intercepts
To find the y-intercept of the line that is midway between the two lines, we take the average of the y-intercepts:
\[ \text{Midpoint} = \frac{3 + \left(-\frac{3}{2}\right)}{2} = \frac{3 - \frac{3}{2}}{2} = \frac{\frac{6}{2} - \frac{3}{2}}{2} = \frac{\frac{3}{2}}{2} = \frac{3}{4} \]

### Step 4: Write the equation of the midway line
Since the midway line is parallel to the original lines, it will have the same slope. The slope from the original lines is \(\frac{3}{4}\). Therefore, the equation of the midway line can be expressed as:
\[ y = \frac{3}{4}x + \frac{3}{4} \]

### Step 5: Convert to standard form
To convert this back to standard form \(Ax + By + C = 0\):
\[ y - \frac{3}{4}x - \frac{3}{4} = 0 \implies \frac{3}{4}x - y + \frac{3}{4} = 0 \]
Multiplying through by \(4\) to eliminate the fraction:
\[ 3x - 4y + 3 = 0 \]

### Final Equation
Thus, the equation of the line midway between the two given lines is:
\[ 3x - 4y + 3 = 0 \]