Find the distance between the parallel lines
3x+4y−7=0and3x+4y+23=0.
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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=0and3x+4y+23=0, we can identify the coefficients A ,B, C1 ,andC2. Here, A=3 ,B=4, C1=−7 ,andC2=23 $
.
Step 2: Apply the distance formula for parallel lines
The distance d between two parallel lines Ax + By + C1=0 and Ax + By + C2=0 is given by the formula d = \frac{|C_1 - C_2|} {A2+B2}$. 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, C1=−7, and C2=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 C1 and C2 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.
32=9and42=16
. So, 9 + 16 = 25
,and25=5
.
Step 6: Find the distance
Finally, we divide the numerator by the denominator to find the distance d
.30÷5=6
. Therefore, the distance between the two parallel lines is 6 units.