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If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3

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Question: If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3, what is the distance between them?

Options:

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Correct Answer: 2

Solution: 

The distance between two parallel lines of the form y = mx + b1 and y = mx + b2 is given by |b2 - b1| / sqrt(1 + m^2). Here, |(-3) - 1| / sqrt(1 + 4) = 4 / sqrt(5) which approximates to 1.

If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3

Practice Questions

Q1
If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3, what is the distance between them?
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Questions & Step-by-Step Solutions

If two parallel lines are represented by the equations y = 4x + 1 and y = 4x - 3, what is the distance between them?
  • Step 1: Identify the equations of the two parallel lines. They are y = 4x + 1 and y = 4x - 3.
  • Step 2: Recognize that both lines have the same slope (m = 4), which means they are parallel.
  • Step 3: Identify the y-intercepts (b1 and b2) of the lines. For the first line, b1 = 1, and for the second line, b2 = -3.
  • Step 4: Use the formula for the distance between two parallel lines: Distance = |b2 - b1| / sqrt(1 + m^2).
  • Step 5: Calculate the difference between the y-intercepts: |(-3) - 1| = |-4| = 4.
  • Step 6: Calculate the value of sqrt(1 + m^2): m = 4, so m^2 = 16, and 1 + m^2 = 17, thus sqrt(17).
  • Step 7: Substitute the values into the distance formula: Distance = 4 / sqrt(17).
  • Step 8: Approximate the distance: 4 / sqrt(17) is approximately 0.97.
  • Distance Between Parallel Lines – The formula for calculating the distance between two parallel lines given in slope-intercept form.
  • Understanding Slope-Intercept Form – Recognizing the structure of the equations of lines in the form y = mx + b.
  • Absolute Value and Square Root – Applying absolute value and square root in the context of the distance formula.
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