The problem asks to calculate the angle between two lines given by the equations:

1. $\frac{-2x}{2} = \frac{y+5}{-3}$
2. $y = 4x$ and $y = 2 - 5x$

Step 1: Simplify the equations of the lines.

Line 1:

Given equation: $\frac{-2x}{2} = \frac{y + 5}{-3}$ This simplifies to: $-x = \frac{y + 5}{-3}$ Multiply both sides by $-3$ to get: $3x = y + 5$ So, the equation of the first line is: $y = 3x - 5$

Line 2:

We have two separate lines given by the equations: $y = 4x \quad \text{and} \quad y = 2 - 5x$ Let's analyze the angles between each pair of lines.

Step 2: Find the slopes of the lines.

• For $y = 3x - 5$, the slope $m_1 = 3$.
• For $y = 4x$, the slope $m_2 = 4$.
• For $y = 2 - 5x$, the slope $m_3 = -5$.

Step 3: Use the formula to calculate the angle between two lines.

The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$

Angle between Line 1 and Line 2 ($y = 3x - 5$ and $y = 4x$):

$\tan \theta_1 = \left|\frac{3 - 4}{1 + 3 \times 4}\right| = \left|\frac{-1}{13}\right| = \frac{1}{13}$

So, $\theta_1 = \arctan\left(\frac{1}{13}\right)$.

Angle between Line 1 and Line 3 ($y = 3x - 5$ and $y = 2 - 5x$):

$\tan \theta_2 = \left|\frac{3 - (-5)}{1 + 3 \times (-5)}\right| = \left|\frac{8}{1 - 15}\right| = \left|\frac{8}{-14}\right| = \frac{4}{7}$

So, $\theta_2 = \arctan\left(\frac{4}{7}\right)$.

Final Answers:

• The angle between the lines $y = 3x - 5$ and $y = 4x$ is approximately $\theta_1 = \arctan\left(\frac{1}{13}\right)$.
• The angle between the lines $y = 3x - 5$ and $y = 2 - 5x$ is approximately $\theta_2 = \arctan\left(\frac{4}{7}\right)$.

Would you like the exact angles in degrees or radians, or any further details?

Related Questions:

1. How can you find the intersection point of two lines?
2. What is the geometric interpretation of the angle between two lines?
3. How does the slope of a line relate to its angle with the x-axis?
4. What is the significance of parallel lines in terms of angles?
5. How do you determine if two lines are perpendicular based on their slopes?

Tip:

Remember that the angle between two perpendicular lines is always 90 degrees, and their slopes are negative reciprocals of each other.