Find the distance between the parallel lines 3x−4y+7=0 and 3x−4y+5=0.
Get the complete, step-by-step math solution for: "Find the distance between the parallel lines 3x - 4y +7 = 0 and 3x - 4y + 5 = 0.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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+C1=0 and Ax+By+C2=0.
Step 2: Extract coefficients and constants
From the given equations, we can identify the values of A, B, C1, and C2. For both lines, A=3 and B=−4. The constant terms are C1=7 for the first line and C2=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+C1=0 and Ax+By+C2=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, C1=7, and C2=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: 32=9 and (−4)2=16, so 9+16=25=5. Finally, we divide the numerator by the denominator to get the distance.