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The problem asks for the equations of parallel ("//") and perpendicular ("⊥") lines to a given set of equations, and then graph them. Let's work through each one step by step.

### The key points to remember:
- **Parallel lines** have the same slope as the given line.
- **Perpendicular lines** have slopes that are the negative reciprocal of the slope of the given line.

For each equation in the list, I'll first determine the slope (from the slope-intercept form \(y = mx + b\), where \(m\) is the slope), then I'll write the equation of the parallel and perpendicular lines.

### 1. \(y_1 = 3x - 4\)
- **Slope** \(m_1 = 3\)
- **Parallel line**: Same slope, so the equation is \(y = 3x + b\), where \(b\) can be any value.
- **Perpendicular line**: Slope is the negative reciprocal of 3, which is \(-\frac{1}{3}\). The equation is \(y = -\frac{1}{3}x + b\), where \(b\) can be any value.

### 2. \(y_2 = -2x - 5\)
- **Slope** \(m_2 = -2\)
- **Parallel line**: Same slope, \(y = -2x + b\).
- **Perpendicular line**: Slope is the negative reciprocal of \(-2\), which is \(\frac{1}{2}\). The equation is \(y = \frac{1}{2}x + b\).

### 3. \(y_3 = x\)
- **Slope** \(m_3 = 1\)
- **Parallel line**: Same slope, \(y = x + b\).
- **Perpendicular line**: Slope is the negative reciprocal of \(1\), which is \(-1\). The equation is \(y = -x + b\).

### 4. \(y_4 = -\frac{2}{5}x - 4\)
- **Slope** \(m_4 = -\frac{2}{5}\)
- **Parallel line**: Same slope, \(y = -\frac{2}{5}x + b\).
- **Perpendicular line**: Slope is the negative reciprocal of \(-\frac{2}{5}\), which is \(\frac{5}{2}\). The equation is \(y = \frac{5}{2}x + b\).

### 5. \(y_5 = -3\)
- This is a horizontal line with **slope** \(m_5 = 0\).
- **Parallel line**: Another horizontal line, so \(y = -3 + b\), which simplifies to \(y = c\), where \(c\) is any constant.
- **Perpendicular line**: A perpendicular to a horizontal line is a vertical line, which has an undefined slope. The equation of a vertical line is \(x = k\), where \(k\) is a constant.

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Would you like to review specific examples in more detail or have more questions? Here are five questions to guide further exploration:

1. How do we find the slope of a line from its equation in general?
2. Why are parallel lines required to have the same slope?
3. How can we graph a vertical line, and why does it have no slope?
4. What happens to the slope of a line when finding a perpendicular line?
5. Can we have multiple parallel or perpendicular lines for a given line?

**Tip**: Always remember that the slope of perpendicular lines are negative reciprocals!

Dar la ecuación de una recta paralela (//) y una perpendicular (⊥) a cada una de las siguientes ecuaciones de rectas. Luego graficarlas. y1 = 3x - 4, y2 = -2x - 5, y3 = x, y4 = -2/5x - 4, y5 = -3

Learn how to find the equations of lines parallel and perpendicular to given linear equations. This problem involves understanding slopes, slope-intercept form, and the properties of parallel and perpendicular lines. Suitable for students in Grades 9-10.

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