If 3y₁ = -4x + 7 and 2y₂ = 6x + 12, find the value of tanθ, where θ is the angle between the lines.

To find the angle between two lines, we use the formula for the tangent of the angle \( \theta \) between two lines given by their slopes \( m_1 \) and \( m_2 \):

\[
\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|
\]

First, let's rewrite the equations of the lines in slope-intercept form \( y = mx + c \) to identify the slopes \( m_1 \) and \( m_2 \).

### Line 1: \( 3y_1 = -4x + 7 \)
Divide both sides by 3:
\[
y_1 = -\frac{4}{3}x + \frac{7}{3}
\]
Thus, the slope of the first line is:
\[
m_1 = -\frac{4}{3}
\]

### Line 2: \( 2y_2 = 6x + 12 \)
Divide both sides by 2:
\[
y_2 = 3x + 6
\]
Thus, the slope of the second line is:
\[
m_2 = 3
\]

### Now, using the formula for \( \tan \theta \):
\[
\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|
\]
Substitute \( m_1 = -\frac{4}{3} \) and \( m_2 = 3 \):
\[
\tan \theta = \left| \frac{-\frac{4}{3} - 3}{1 + \left( -\frac{4}{3} \times 3 \right)} \right|
\]
Simplify:
\[
\tan \theta = \left| \frac{-\frac{4}{3} - \frac{9}{3}}{1 + (-4)} \right| = \left| \frac{-\frac{13}{3}}{-3} \right|
\]
\[
\tan \theta = \left| \frac{-13}{3} \times \frac{1}{-3} \right| = \left| \frac{13}{9} \right| = \frac{13}{9}
\]

Thus, the value of \( \tan \theta \) is \( \frac{13}{9} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the relationship between the slopes of two perpendicular lines?
2. How do you find the angle between two perpendicular lines?
3. How would you modify this process if the lines were parallel?
4. What is the geometric significance of the tangent function in terms of angle?
5. How can you find the angle if only the direction vectors of the lines are given?

**Tip:** The tangent of the angle between two lines helps determine whether the lines are parallel, perpendicular, or form any other specific angle.

Explore how to find the value of tanθ for the angle between two lines represented by the equations 3y₁ = -4x + 7 and 2y₂ = 6x + 12. This solution uses slope-intercept form and the tangent angle formula. Suitable for students in Grades 10-12.

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