Find the distance between the parallel lines $3x - 4y +7 = 0$ and $3x - 4y + 5 = 0$.

Step 1: Identify the general form of parallel lines

The given equations of the lines are $3x - 4y + 7 = 0$ and $3x - 4y + 5 = 0$. We can see that the coefficients of $x$ and $y$ are the same for both equations, which confirms that the lines are parallel. We can write them in the general form $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$.

Step 2: Extract coefficients and constants

From the given equations, we can identify the values of $A$, $B$, $C_1$, and $C_2$. For both lines, $A = 3$ and $B = -4$. The constant terms are $C_1 = 7$ for the first line and $C_2 = 5$ for the second line.

Step 3: Apply the distance formula for parallel lines

The formula to find the perpendicular distance $d$ between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by the absolute difference of the constant terms divided by the square root of the sum of the squares of the coefficients of $x$ and $y$.

Step 4: Substitute the values into the formula

Now, we substitute the values of $A=3$, $B=-4$, $C_1=7$, and $C_2=5$ into the distance formula. This will allow us to calculate the distance between the two parallel lines.

Step 5: Calculate the distance

First, we calculate the absolute difference in the numerator, which is $|7 - 5| = 2$. Then, we calculate the denominator: $3^2 = 9$ and $(-4)^2 = 16$, so $\sqrt{9 + 16} = \sqrt{25} = 5$. Finally, we divide the numerator by the denominator to get the distance.

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