Find the distance between the parallel lines 4x-3y+5=0 and 4x-3y=0

Answer

To find the distance between the parallel lines given by the equations \(4x - 3y + 5 = 0\) and \(4x - 3y = 0\), we can use the formula for the distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\):

\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]

### Step-by-Step Solution:

1. **Identify the coefficients and constants from the equations**:
- For the first line \(4x - 3y + 5 = 0\):
- \(A = 4\), \(B = -3\), \(C_1 = 5\)
- For the second line \(4x - 3y = 0\):
- \(A = 4\), \(B = -3\), \(C_2 = 0\)

2. **Substitute the values into the distance formula**:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} = \frac{|5 - 0|}{\sqrt{4^2 + (-3)^2}} \]

3. **Calculate the numerator**:
\[ |C_1 - C_2| = |5 - 0| = 5 \]

4. **Calculate the denominator**:
\[ \sqrt{A^2 + B^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

5. **Calculate the distance**:
\[ d = \frac{5}{5} = 1 \]

### Final Answer:
The distance between the parallel lines \(4x - 3y + 5 = 0\) and \(4x - 3y = 0\) is \(1\) unit.

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