What is the angle between the lines 2x + 3y = 6 and 4x - y = 5?

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Question: What is the angle between the lines 2x + 3y = 6 and 4x - y = 5?

Options:

Correct Answer: 60 degrees

Solution:

The slopes of the lines are -2/3 and 4. The angle θ can be found using tan(θ) = |(m1 - m2) / (1 + m1*m2)|.

What is the angle between the lines 2x + 3y = 6 and 4x - y = 5?

What is the angle between the lines 2x + 3y = 6 and 4x - y = 5?

Step 1: Write down the equations of the lines: Line 1 is 2x + 3y = 6 and Line 2 is 4x - y = 5.
Step 2: Rearrange Line 1 (2x + 3y = 6) to the slope-intercept form (y = mx + b).
Step 3: Solve for y in Line 1: 3y = -2x + 6, then y = -2/3x + 2. The slope (m1) is -2/3.
Step 4: Rearrange Line 2 (4x - y = 5) to the slope-intercept form (y = mx + b).
Step 5: Solve for y in Line 2: -y = -4x + 5, then y = 4x - 5. The slope (m2) is 4.
Step 6: Use the formula for the angle θ between two lines: tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Step 7: Substitute m1 = -2/3 and m2 = 4 into the formula: tan(θ) = |(-2/3 - 4) / (1 + (-2/3)*4)|.
Step 8: Calculate the numerator: -2/3 - 4 = -2/3 - 12/3 = -14/3.
Step 9: Calculate the denominator: 1 + (-2/3)*4 = 1 - 8/3 = 3/3 - 8/3 = -5/3.
Step 10: Now, tan(θ) = |(-14/3) / (-5/3)| = |14/5|.
Step 11: Find θ by taking the arctan of 14/5.

Finding the slope of a line – Understanding how to convert the equation of a line from standard form to slope-intercept form to determine its slope.
Angle between two lines – Using the formula for the angle between two lines based on their slopes to find the angle in degrees or radians.
Trigonometric functions – Applying the tangent function to calculate the angle based on the difference and product of the slopes.