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The task is to find the angle between two lines. We are given:

1. **Line A**: \( 2x - 3y + 4 = 0 \)
2. **Line B**: \( y = -4x - 9 \)

### Step 1: Convert both equations to slope-intercept form (\( y = mx + b \)) to identify their slopes.

- **Line A**: Rearrange \( 2x - 3y + 4 = 0 \) to solve for \( y \).
  \[
  2x - 3y + 4 = 0 \implies -3y = -2x - 4 \implies y = \frac{2}{3}x + \frac{4}{3}
  \]
  So, the slope of Line A is \( m_A = \frac{2}{3} \).

- **Line B** is already in slope-intercept form \( y = -4x - 9 \), so the slope of Line B is \( m_B = -4 \).

### Step 2: Use the formula to find the angle \( \theta \) between two lines:
\[
\tan(\theta) = \left| \frac{m_A - m_B}{1 + m_A m_B} \right|
\]
Substitute \( m_A = \frac{2}{3} \) and \( m_B = -4 \):
\[
\tan(\theta) = \left| \frac{\frac{2}{3} - (-4)}{1 + \frac{2}{3} \times (-4)} \right| = \left| \frac{\frac{2}{3} + 4}{1 - \frac{8}{3}} \right| = \left| \frac{\frac{2}{3} + \frac{12}{3}}{1 - \frac{8}{3}} \right| = \left| \frac{\frac{14}{3}}{\frac{-5}{3}} \right| = \left| -\frac{14}{5} \right|
\]
Thus,
\[
\tan(\theta) = \frac{14}{5}
\]

### Step 3: Find the angle \( \theta \) by taking the arctangent:
\[
\theta = \tan^{-1}\left( \frac{14}{5} \right)
\]

Let me calculate this for you.The angle between the two lines is approximately \( 70.35^\circ \).

Would you like further clarification or more details on this? Here are five related questions you might find interesting:

1. How do you find the angle between two parallel lines?
2. What happens if the slopes of two lines are equal?
3. How do you calculate the angle between two lines in 3D space?
4. What is the relationship between the slopes of perpendicular lines?
5. How does changing the slopes affect the angle between two lines?

**Tip:** When calculating angles between lines, ensure that the slopes are accurately converted into their proper forms, especially when rearranging equations.

Encuentra el ángulo entre el par de líneas: 2x - 3y + 4 = 0 y y = -4x - 9.

Learn how to calculate the angle between two lines using the slopes of 2x - 3y + 4 = 0 and y = -4x - 9. This detailed solution walks through the conversion of equations into slope-intercept form, application of the slope-angle formula, and final trigonometric calculation. Suitable for students in Grades 9-12.

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