If two parallel lines are represented by the equations y = 3x + 2 and y = 3x - 4

If two parallel lines are represented by the equations y = 3x + 2 and y = 3x - 4, what is the distance between these two lines?

If two parallel lines are represented by the equations y = 3x + 2 and y = 3x - 4, what is the distance between these two lines?

Step 1: Identify the equations of the two parallel lines. They are y = 3x + 2 and y = 3x - 4.
Step 2: Recognize that the lines are in the form y = mx + b, where m is the slope and b is the y-intercept.
Step 3: Note that both lines have the same slope (m = 3), which confirms they are parallel.
Step 4: Identify the y-intercepts (b1 and b2) from the equations. Here, b1 = 2 and b2 = -4.
Step 5: Calculate the absolute difference between the y-intercepts: |b2 - b1| = |-4 - 2| = |-6| = 6.
Step 6: Calculate the value of √(1 + m^2). Since m = 3, we find √(1 + 3^2) = √(1 + 9) = √10.
Step 7: Use the formula for the distance between the two parallel lines: Distance = |b2 - b1| / √(1 + m^2). Substitute the values: Distance = 6 / √10.
Step 8: The final answer is the distance between the two lines, which is 6 / √10.

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