Question: In Problems 37-40, find the slope and y-intercept of each line. Graph the line, labeling any intercepts.
37. 4x - 5y = -20
38. 3x + 4y = 12
39. 1/2 x - 1/3 y = -1/6
40. - 3/4 x + 1/2 y = 0
In Problems 41-44, find the intercepts and graph each line.
41. 2x - 3y = 12
42. x - 2y = 8
43. 1/2 x + 1/3 y = 2
44. 1/3 x - 1/4 y = 1
Position hints:
39: top-center
42: bottom-center
1. **Problem Statement:** Find the slope and y-intercept for lines 37-40, then find intercepts for lines 41-44.
---
### Problems 37-40: Find slope and y-intercept
**General form:** $$Ax + By = C$$
**Slope-intercept form:** $$y = mx + b$$ where $$m = -\frac{A}{B}$$ and $$b = \frac{C}{B}$$
---
**37.** $$4x - 5y = -20$$
1. Solve for $$y$$:
$$-5y = -4x - 20$$
$$y = \frac{-4x - 20}{-5} = \frac{-4x}{-5} + \frac{-20}{-5} = \frac{4}{5}x + 4$$
2. Slope $$m = \frac{4}{5}$$, y-intercept $$b = 4$$
---
**38.** $$3x + 4y = 12$$
1. Solve for $$y$$:
$$4y = -3x + 12$$
$$y = \frac{-3x + 12}{4} = -\frac{3}{4}x + 3$$
2. Slope $$m = -\frac{3}{4}$$, y-intercept $$b = 3$$
---
**39.** $$\frac{1}{2}x - \frac{1}{3}y = -\frac{1}{6}$$
1. Multiply entire equation by 6 to clear denominators:
$$6 \times \left(\frac{1}{2}x - \frac{1}{3}y\right) = 6 \times -\frac{1}{6}$$
$$3x - 2y = -1$$
2. Solve for $$y$$:
$$-2y = -3x - 1$$
$$y = \frac{-3x - 1}{-2} = \frac{-3x}{-2} + \frac{-1}{-2} = \frac{3}{2}x + \frac{1}{2}$$
3. Slope $$m = \frac{3}{2}$$, y-intercept $$b = \frac{1}{2}$$
---
**40.** $$- \frac{3}{4}x + \frac{1}{2}y = 0$$
1. Multiply entire equation by 4 to clear denominators:
$$4 \times \left(- \frac{3}{4}x + \frac{1}{2}y\right) = 4 \times 0$$
$$-3x + 2y = 0$$
2. Solve for $$y$$:
$$2y = 3x$$
$$y = \frac{3}{2}x$$
3. Slope $$m = \frac{3}{2}$$, y-intercept $$b = 0$$
---
### Problems 41-44: Find intercepts
**Intercepts:**
- x-intercept: set $$y=0$$ and solve for $$x$$
- y-intercept: set $$x=0$$ and solve for $$y$$
---
**41.** $$2x - 3y = 12$$
- x-intercept: $$2x - 3(0) = 12 \Rightarrow 2x = 12 \Rightarrow x = 6$$
- y-intercept: $$2(0) - 3y = 12 \Rightarrow -3y = 12 \Rightarrow y = -4$$
---
**42.** $$x - 2y = 8$$
- x-intercept: $$x - 2(0) = 8 \Rightarrow x = 8$$
- y-intercept: $$0 - 2y = 8 \Rightarrow -2y = 8 \Rightarrow y = -4$$
---
**43.** $$\frac{1}{2}x + \frac{1}{3}y = 2$$
1. Multiply entire equation by 6:
$$6 \times \left(\frac{1}{2}x + \frac{1}{3}y\right) = 6 \times 2$$
$$3x + 2y = 12$$
- x-intercept: $$3x + 2(0) = 12 \Rightarrow 3x = 12 \Rightarrow x = 4$$
- y-intercept: $$3(0) + 2y = 12 \Rightarrow 2y = 12 \Rightarrow y = 6$$
---
**44.** $$\frac{1}{3}x - \frac{1}{4}y = 1$$
1. Multiply entire equation by 12:
$$12 \times \left(\frac{1}{3}x - \frac{1}{4}y\right) = 12 \times 1$$
$$4x - 3y = 12$$
- x-intercept: $$4x - 3(0) = 12 \Rightarrow 4x = 12 \Rightarrow x = 3$$
- y-intercept: $$4(0) - 3y = 12 \Rightarrow -3y = 12 \Rightarrow y = -4$$