What is the angle between the lines represented by the equations y = 2x + 1 and

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Question: What is the angle between the lines represented by the equations y = 2x + 1 and y = -1/2x + 3? (2021)

Options:

Correct Answer: 90 degrees

Exam Year: 2021

Solution:

The slopes are m1 = 2 and m2 = -1/2. The angle θ between the lines is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)|, which results in 90 degrees.

What is the angle between the lines represented by the equations y = 2x + 1 and y = -1/2x + 3? (2021)

What is the angle between the lines represented by the equations y = 2x + 1 and y = -1/2x + 3? (2021)

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: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Step 5: Substitute the values of m1 and m2 into the formula: tan(θ) = |(2 - (-1/2)) / (1 + 2 * (-1/2))|.
Step 6: Simplify the expression: tan(θ) = |(2 + 1/2) / (1 - 1)|. Note that the denominator becomes 0.
Step 7: Since the denominator is 0, this indicates that the lines are perpendicular, which means the angle θ is 90 degrees.

Slope of a Line – Understanding how to find the slope from the equation of a line in slope-intercept form (y = mx + b).
Angle Between Two Lines – Using the formula for the angle between two lines based on their slopes.
Trigonometric Functions – Applying the tangent function to find the angle between two lines.