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### 2) Write the slope-intercept form of the equation of the line through the given point with the given slope.

**Given:**  
- Point: \((-1, -2)\)  
- Slope: undefined

**Explanation:**  
An undefined slope implies a vertical line. The equation of a vertical line through the point \((-1, -2)\) is of the form:
\[ x = -1 \]
This is the equation of the line.

---

### 3) Write the slope-intercept form of the equation of the line through the given points.

**Given Points:**  
- \((3, 3)\) and \((-2, 0)\)

**Step 1: Find the slope \(m\)**  
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{-2 - 3} = \frac{-3}{-5} = \frac{3}{5}
\]
So, the slope is \( \frac{3}{5} \).

**Step 2: Use the point-slope form**  
We can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((3, 3)\) and \(m = \frac{3}{5}\):
\[
y - 3 = \frac{3}{5}(x - 3)
\]
Expanding this:
\[
y - 3 = \frac{3}{5}x - \frac{9}{5}
\]
\[
y = \frac{3}{5}x - \frac{9}{5} + \frac{15}{5}
\]
\[
y = \frac{3}{5}x + \frac{6}{5}
\]
This is the slope-intercept form of the equation.

---

### 4) Write the slope-intercept form of the equation of the line described.

**Given:**  
- Point: \((4, -1)\)
- Parallel to \(y = x - 1\)

**Step 1: Identify the slope**  
The slope of the line \(y = x - 1\) is \(m = 1\) because it's in the form \(y = mx + b\).

**Step 2: Use point-slope form**  
Using the point \((4, -1)\) and slope \(m = 1\):
\[
y - (-1) = 1(x - 4)
\]
Simplify:
\[
y + 1 = x - 4
\]
\[
y = x - 5
\]
So, the slope-intercept form of the equation is:
\[
y = x - 5
\]

---

### 5) Write the slope-intercept form of the equation of the line described.

**Given:**  
- Point: \((2, 5)\)
- Perpendicular to \(y = -2x - 4\)

**Step 1: Identify the slope**  
The slope of the line \(y = -2x - 4\) is \(m = -2\). The slope of a line perpendicular to this is the negative reciprocal, so the new slope will be:
\[
m = \frac{1}{2}
\]

**Step 2: Use point-slope form**  
Using the point \((2, 5)\) and slope \(m = \frac{1}{2}\):
\[
y - 5 = \frac{1}{2}(x - 2)
\]
Expanding:
\[
y - 5 = \frac{1}{2}x - 1
\]
\[
y = \frac{1}{2}x - 1 + 5
\]
\[
y = \frac{1}{2}x + 4
\]
This is the slope-intercept form of the equation.

---

### 6) Write the standard form of the equation of each line.

**Given Line:**  
\[
y = \frac{1}{3}x + 2
\]

**Convert to standard form \(Ax + By = C\):**

Multiply through by 3 to eliminate the fraction:
\[
3y = x + 6
\]
Rearrange:
\[
x - 3y = -6
\]
This is the standard form of the equation:  
\[
x - 3y = -6
\]

---

Do you need further clarifications or details on any part? Here are 5 related questions for more practice:

1. What is the equation of a line with an undefined slope passing through the point \((5, 1)\)?
2. Find the equation of the line passing through the points \((1, 2)\) and \((4, 8)\).
3. What is the equation of a line perpendicular to \(y = \frac{1}{4}x + 3\) passing through the point \((0, -1)\)?
4. Convert the equation \(y = 2x + 5\) to standard form.
5. What is the slope of a line parallel to \(y = -\frac{3}{4}x + 2\)?

**Tip:**  
For parallel lines, always remember that they share the same slope!

Write the slope-intercept form of the equation of the line through given points, find parallel and perpendicular slopes, and convert to standard form.

Learn how to write equations in slope-intercept and standard form through given points, find parallel and perpendicular slopes, and convert forms. Problems include finding the equation of lines through (3, 3) and (-2, 0) and calculating the line parallel to y = x - 1.

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