What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

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Question: What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

Options:

Correct Answer: 60 degrees

Solution:

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.

What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

Step 1: Identify the equations of the lines. The first line is y = 2x + 1 and the second line is y = -1/2x + 3.
Step 2: Find the slope of the first line (m1). The slope is the coefficient of x, which is 2.
Step 3: Find the slope of the second line (m2). The slope is the coefficient of x, which is -1/2.
Step 4: Use the formula to find the angle θ between the two lines: θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)|.
Step 5: Substitute the values of m1 and m2 into the formula: θ = tan^(-1) |(2 - (-1/2)) / (1 + 2 * (-1/2))|.
Step 6: Simplify the expression: (2 + 1/2) = 5/2 and (1 - 1) = 0, so we have θ = tan^(-1)(5/4).
Step 7: Calculate θ using a calculator or trigonometric tables to find that θ is approximately 60 degrees.

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