Math Problem Statement

The equation 2x + 3y + 4 = 0 makes an angle with the positive x-axis. Which of the following is closest to the size of the angle? A. 34° B. 56° C. 124° D. 146°

Solution

To solve the problem, we need to find the angle that the line 2x+3y+4=02x + 3y + 4 = 0 makes with the positive x-axis.

Steps:

  1. Find the slope of the line: The equation of the line is given in general form Ax+By+C=0Ax + By + C = 0. To convert it to slope-intercept form y=mx+by = mx + b, solve for yy:

    2x+3y+4=02x + 3y + 4 = 0 3y=2x43y = -2x - 4 y=23x43y = -\frac{2}{3}x - \frac{4}{3}

    The slope mm is 23-\frac{2}{3}.

  2. Use the slope to find the angle: The angle θ\theta that the line makes with the positive x-axis can be found using the formula: tan(θ)=m\tan(\theta) = |m| tan(θ)=23=23\tan(\theta) = \left| -\frac{2}{3} \right| = \frac{2}{3}

    Taking the arctangent of both sides: θ=tan1(23)\theta = \tan^{-1}\left(\frac{2}{3}\right)

  3. Calculate the angle:

    Let me compute this.

The angle θ\theta that the line makes with the positive x-axis is approximately 33.6933.69^\circ, which is closest to option A: 3434^\circ.

Would you like more details on this solution or have any other questions?

Here are five related questions to explore:

  1. How do you derive the slope-intercept form from the general form of a line?
  2. What is the relationship between slope and angle for a line?
  3. How can you calculate the angle if the slope is negative or positive?
  4. Why is the arctangent function used to find the angle from the slope?
  5. How would the answer change if the slope were steeper?

Tip: For any line, the angle with the x-axis can be found directly from its slope using the inverse tangent function.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Trigonometry

Formulas

Slope of a line: m = -A/B
tan(θ) = |m|
θ = arctan(m)

Theorems

Slope-Intercept Form Theorem
Inverse Tangent Function

Suitable Grade Level

Grades 9-12