Find the distance between the parallel lines $$3x + 4y - 7 = 0 and 3x + 4y + 23 = 0.$$

Step 1: Identify the coefficients

We are given two parallel lines in the standard form Ax + By + $C = 0$ From the equations $$3x + 4y - 7 = 0 and 3x + 4y + 23 = 0$$ , we can identify the coefficients A $$, B$$ , $C_1$ $$, and C_2$$ . Here, $A=3$ $$, B=4$$ , $C_1=-7$ $$, and C_2=23$$ .

Step 2: Apply the distance formula for parallel lines

The distance d between two parallel lines Ax + By + $C_1 = 0$ and Ax + By + $C_2 = 0$ is given by the formula d = \frac{|C_1 - C_2|} {$\sqrt{A^2 + B^2}$}. This formula calculates the perpendicular distance between the two lines.

Step 3: Substitute the values

Now, we substitute the identified values of $A=3$ $$, B=4$$ , $C_1=-7$, and $C_2=23$ into the distance formula. This will allow us to calculate the numerical distance.

Step 4: Calculate the numerator

First, we calculate the numerator of the formula. The difference between $C_1$ and $C_2$ is -7 - 23 = -30 Taking the absolute value, we get $$|-30| = 30$$ .

Step 5: Calculate the denominator

Next, we calculate the denominator. We square A and B add them, and then take the square root. $$3^2 = 9 and 4^2 = 16$$ . So, 9 + 16 = 25 $$, and \sqrt{25} = 5$$ .

Step 6: Find the distance

Finally, we divide the numerator by the denominator to find the distance d $$. 30 \div 5 = 6$$ . Therefore, the distance between the two parallel lines is 6 units.

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