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

Get the complete, step-by-step math solution for: "Find the distance between the parallel lines 3x + 4y - 7 = 0 and 3x + 4y + 23 = 0....". Powered by SolveForX AI math tutor.

Step-by-Step Solution

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.