If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5

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Question: If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5, what is the distance between them?

Options:

Correct Answer: 8/√5

Solution:

The distance between two parallel lines y = mx + b1 and y = mx + b2 is |b2 - b1| / √(1 + m^2). Here, |(-5) - 3| / √(1 + 2^2) = 8/√5.

If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5, what is the distance between them?

If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5, what is the distance between them?

Step 1: Identify the equations of the two parallel lines. They are y = 2x + 3 and y = 2x - 5.
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 = 2), which confirms they are parallel.
Step 4: Identify the y-intercepts (b1 and b2) from the equations. Here, b1 = 3 and b2 = -5.
Step 5: Use the formula for the distance between two parallel lines: Distance = |b2 - b1| / √(1 + m^2).
Step 6: Calculate |b2 - b1|: |(-5) - 3| = |-8| = 8.
Step 7: Calculate m^2: m^2 = 2^2 = 4, so 1 + m^2 = 1 + 4 = 5.
Step 8: Calculate the square root: √(5).
Step 9: Substitute the values into the distance formula: Distance = 8 / √(5).
Step 10: The final answer for the distance between the two parallel lines is 8 / √(5).

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 linear equations in the form y = mx + b, where m is the slope and b is the y-intercept.