https://file.aiquickdraw.com/mathbot/file/17291958758668l0oj00o.jpg

Let's break down the two problems from the image.

### 5) Determine which lines, if any, are parallel or perpendicular.

#### Given lines:
- **Line a:** \( y = \frac{3}{5}x + 1 \) (already in slope-intercept form \( y = mx + b \))
- **Line b:** \( 5y = 3x - 2 \) → Rewrite in slope-intercept form:
  \[
  y = \frac{3}{5}x - \frac{2}{5}
  \]
- **Line c:** \( 10x - 6y = -4 \) → Rewrite in slope-intercept form:
  \[
  6y = 10x + 4 \quad \Rightarrow \quad y = \frac{5}{3}x + \frac{2}{3}
  \]

#### Slope comparison:
- **Line a:** slope \( m = \frac{3}{5} \)
- **Line b:** slope \( m = \frac{3}{5} \)
- **Line c:** slope \( m = \frac{5}{3} \)

#### Conclusion:
- **Lines a and b** are parallel because they have the same slope (\( \frac{3}{5} \)).
- **Line c** is **perpendicular** to both lines a and b because the slope of line c (\( \frac{5}{3} \)) is the negative reciprocal of the slope of lines a and b (\( \frac{3}{5} \)).

### 6) Determine which lines, if any, are parallel or perpendicular.

#### Given lines:
- **Line a:** \( 4x - 3y = 2 \) → Rewrite in slope-intercept form:
  \[
  3y = 4x - 2 \quad \Rightarrow \quad y = \frac{4}{3}x - \frac{2}{3}
  \]
- **Line b:** \( 3x + 4y = -1 \) → Rewrite in slope-intercept form:
  \[
  4y = -3x - 1 \quad \Rightarrow \quad y = -\frac{3}{4}x - \frac{1}{4}
  \]
- **Line c:** \( 4y - 3x = 20 \) → Rewrite in slope-intercept form:
  \[
  4y = 3x + 20 \quad \Rightarrow \quad y = \frac{3}{4}x + 5
  \]

#### Slope comparison:
- **Line a:** slope \( m = \frac{4}{3} \)
- **Line b:** slope \( m = -\frac{3}{4} \)
- **Line c:** slope \( m = \frac{3}{4} \)

#### Conclusion:
- **Line b** is **perpendicular** to both lines a and c, since its slope (\( -\frac{3}{4} \)) is the negative reciprocal of both their slopes.
- **Lines b and c** are neither parallel nor perpendicular, since their slopes are not the same, nor are they negative reciprocals.

Would you like more details or have any questions?

Here are some related questions:
1. How do you convert equations from standard form to slope-intercept form?
2. What is the geometric interpretation of parallel and perpendicular lines?
3. Can two lines with different y-intercepts be parallel?
4. What happens if two lines have slopes that are negative reciprocals but do not intersect?
5. How does the slope of a line affect its angle with the x-axis?

**Tip:** The negative reciprocal rule is a quick way to check if two lines are perpendicular!

Determine which lines, if any, are parallel or perpendicular in the given equations.

Learn how to determine if lines are parallel or perpendicular based on their equations. This problem involves converting standard form to slope-intercept form and comparing slopes for algebraic analysis. Suitable for students in Grades 9-12.

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