The image contains the following questions:

1. Find the equation of a line passing through the point (-4, 3) with slope $\frac{1}{2}$.
2. Write the equation of the line passing through the points (1, -1) and (3, 5).
3. Find the angle between the lines obtained in (i) and (ii).

Solutions:

(i) Equation of a line through (-4, 3) with slope $\frac{1}{2}$:

The slope-intercept form of a line is: $y - y_1 = m(x - x_1)$ Substitute $m = \frac{1}{2}$, $x_1 = -4$, and $y_1 = 3$: $y - 3 = \frac{1}{2}(x + 4)$ $y - 3 = \frac{1}{2}x + 2$ $y = \frac{1}{2}x + 5$

The equation of the line is: $y = \frac{1}{2}x + 5$

(ii) Equation of the line passing through (1, -1) and (3, 5):

First, calculate the slope ($m$) of the line: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{3 - 1} = \frac{6}{2} = 3$ Using the slope-intercept form, take one point, say (1, -1), and substitute $m = 3$, $x_1 = 1$, $y_1 = -1$: $y - (-1) = 3(x - 1)$ $y + 1 = 3x - 3$ $y = 3x - 4$

The equation of the line is: $y = 3x - 4$

(iii) Angle between the lines:

The slopes of the lines are:

• Line (i): $m_1 = \frac{1}{2}$
• Line (ii): $m_2 = 3$

The formula for the angle $\theta$ between two lines is: $\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right|$ Substitute $m_1 = \frac{1}{2}$ and $m_2 = 3$: $\tan\theta = \left| \frac{3 - \frac{1}{2}}{1 + \frac{1}{2} \cdot 3} \right|$ $\tan\theta = \left| \frac{\frac{6}{2} - \frac{1}{2}}{1 + \frac{3}{2}} \right|$ $\tan\theta = \left| \frac{\frac{5}{2}}{\frac{5}{2}} \right| = 1$ $\theta = \tan^{-1}(1) = 45^\circ$

The angle between the lines is: $45^\circ$

Summary of Answers:

1. Equation of line: $y = \frac{1}{2}x + 5$
2. Equation of line: $y = 3x - 4$
3. Angle between lines: $45^\circ$

Let me know if you want further explanation or have questions! Here are related questions to deepen understanding:

1. What are the general steps to derive the slope-intercept form of a line?
2. How is the slope calculated for two given points?
3. What are the special cases when the angle between two lines is $90^\circ$ or $0^\circ$?
4. How would the equations change if the points or slope were different?
5. Can the angle between two lines exceed $90^\circ$? Why or why not?

Tip: When working with angles between lines, remember that the slopes must be consistent with the orientation of the lines.