Algebra worksheet for writing equations in slope-intercept form, featuring problems with points, slopes, graphs, and a real-world investing scenario.

Problem Analysis:

The task involves writing equations of lines in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept. The problems are divided into two main sections:

1. Writing equations given a point and a slope.
2. Writing equations given two points.
3. A real-world application involving linear equations.

Let's solve each section step by step.

---

Section 1: Writing Equations Given a Point and a Slope

The general form of the equation of a line is:
\[
y = mx + b
\]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept.

If a point \((x_1, y_1)\) and the slope \( m \) are given, we can use the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
and then rearrange it into slope-intercept form.

#### Problem 1:
- Point: \((-1, 4)\)
- Slope: \( m = -3 \)

Using the point-slope form:
\[
y - 4 = -3(x - (-1))
\]
\[
y - 4 = -3(x + 1)
\]
\[
y - 4 = -3x - 3
\]
\[
y = -3x - 3 + 4
\]
\[
y = -3x + 1
\]

Answer:
\[
\boxed{y = -3x + 1}
\]

#### Problem 2:
- Point: \((4, 1)\)
- Slope: \( m = 1 \)

Using the point-slope form:
\[
y - 1 = 1(x - 4)
\]
\[
y - 1 = x - 4
\]
\[
y = x - 4 + 1
\]
\[
y = x - 3
\]

Answer:
\[
\boxed{y = x - 3}
\]

#### Problem 3:
- Point: \((-1, 2)\)
- Slope: \( m = 2 \)

Using the point-slope form:
\[
y - 2 = 2(x - (-1))
\]
\[
y - 2 = 2(x + 1)
\]
\[
y - 2 = 2x + 2
\]
\[
y = 2x + 2 + 2
\]
\[
y = 2x + 4
\]

Answer:
\[
\boxed{y = 2x + 4}
\]

#### Problem 4:
- Point: \((1, 9)\)
- Slope: \( m = 4 \)

Using the point-slope form:
\[
y - 9 = 4(x - 1)
\]
\[
y - 9 = 4x - 4
\]
\[
y = 4x - 4 + 9
\]
\[
y = 4x + 5
\]

Answer:
\[
\boxed{y = 4x + 5}
\]

#### Problem 5:
- Point: \((4, 2)\)
- Slope: \( m = -2 \)

Using the point-slope form:
\[
y - 2 = -2(x - 4)
\]
\[
y - 2 = -2x + 8
\]
\[
y = -2x + 8 + 2
\]
\[
y = -2x + 10
\]

Answer:
\[
\boxed{y = -2x + 10}
\]

#### Problem 6:
- Point: \((2, -2)\)
- Slope: \( m = 3 \)

Using the point-slope form:
\[
y - (-2) = 3(x - 2)
\]
\[
y + 2 = 3(x - 2)
\]
\[
y + 2 = 3x - 6
\]
\[
y = 3x - 6 - 2
\]
\[
y = 3x - 8
\]

Answer:
\[
\boxed{y = 3x - 8}
\]

#### Problem 7:
- Point: \((3, 0)\)
- Slope: \( m = 5 \)

Using the point-slope form:
\[
y - 0 = 5(x - 3)
\]
\[
y = 5(x - 3)
\]
\[
y = 5x - 15
\]

Answer:
\[
\boxed{y = 5x - 15}
\]

#### Problem 8:
- Point: \((-3, -2)\)
- Slope: \( m = 2 \)

Using the point-slope form:
\[
y - (-2) = 2(x - (-3))
\]
\[
y + 2 = 2(x + 3)
\]
\[
y + 2 = 2x + 6
\]
\[
y = 2x + 6 - 2
\]
\[
y = 2x + 4
\]

Answer:
\[
\boxed{y = 2x + 4}
\]

#### Problem 9:
- Point: \((-5, 4)\)
- Slope: \( m = -4 \)

Using the point-slope form:
\[
y - 4 = -4(x - (-5))
\]
\[
y - 4 = -4(x + 5)
\]
\[
y - 4 = -4x - 20
\]
\[
y = -4x - 20 + 4
\]
\[
y = -4x - 16
\]

Answer:
\[
\boxed{y = -4x - 16}
\]

---

Section 2: Writing Equations Given Two Points

To find the equation of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\):
1. Calculate the slope \( m \) using:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
2. Use the point-slope form to write the equation:
\[
y - y_1 = m(x - x_1)
\]
3. Rearrange into slope-intercept form.

#### Problem 10:
- Points: \((-2, 3)\) and \((3, -2)\)

Calculate the slope:
\[
m = \frac{-2 - 3}{3 - (-2)} = \frac{-2 - 3}{3 + 2} = \frac{-5}{5} = -1
\]

Use the point-slope form with \((-2, 3)\):
\[
y - 3 = -1(x - (-2))
\]
\[
y - 3 = -1(x + 2)
\]
\[
y - 3 = -x - 2
\]
\[
y = -x - 2 + 3
\]
\[
y = -x + 1
\]

Answer:
\[
\boxed{y = -x + 1}
\]

#### Problem 11:
- Points: \((-1, -3)\) and \((1, 1)\)

Calculate the slope:
\[
m = \frac{1 - (-3)}{1 - (-1)} = \frac{1 + 3}{1 + 1} = \frac{4}{2} = 2
\]

Use the point-slope form with \((-1, -3)\):
\[
y - (-3) = 2(x - (-1))
\]
\[
y + 3 = 2(x + 1)
\]
\[
y + 3 = 2x + 2
\]
\[
y = 2x + 2 - 3
\]
\[
y = 2x - 1
\]

Answer:
\[
\boxed{y = 2x - 1}
\]

#### Problem 12:
- Points: \((0, 3)\) and \((2, -1)\)

Calculate the slope:
\[
m = \frac{-1 - 3}{2 - 0} = \frac{-1 - 3}{2} = \frac{-4}{2} = -2
\]

Use the point-slope form with \((0, 3)\):
\[
y - 3 = -2(x - 0)
\]
\[
y - 3 = -2x
\]
\[
y = -2x + 3
\]

Answer:
\[
\boxed{y = -2x + 3}
\]

#### Problem 13:
- Points: \((1, 3)\) and \((-3, -5)\)

Calculate the slope:
\[
m = \frac{-5 - 3}{-3 - 1} = \frac{-5 - 3}{-4} = \frac{-8}{-4} = 2
\]

Use the point-slope form with \((1, 3)\):
\[
y - 3 = 2(x - 1)
\]
\[
y - 3 = 2x - 2
\]
\[
y = 2x - 2 + 3
\]
\[
y = 2x + 1
\]

Answer:
\[
\boxed{y = 2x + 1}
\]

#### Problem 14:
- Points: \((1, 4)\) and \((6, -1)\)

Calculate the slope:
\[
m = \frac{-1 - 4}{6 - 1} = \frac{-1 - 4}{5} = \frac{-5}{5} = -1
\]

Use the point-slope form with \((1, 4)\):
\[
y - 4 = -1(x - 1)
\]
\[
y - 4 = -x + 1
\]
\[
y = -x + 1 + 4
\]
\[
y = -x + 5
\]

Answer:
\[
\boxed{y = -x + 5}
\]

#### Problem 15:
- Points: \((1, -1)\) and \((3, 5)\)

Calculate the slope:
\[
m = \frac{5 - (-1)}{3 - 1} = \frac{5 + 1}{2} = \frac{6}{2} = 3
\]

Use the point-slope form with \((1, -1)\):
\[
y - (-1) = 3(x - 1)
\]
\[
y + 1 = 3(x - 1)
\]
\[
y + 1 = 3x - 3
\]
\[
y = 3x - 3 - 1
\]
\[
y = 3x - 4
\]

Answer:
\[
\boxed{y = 3x - 4}
\]

#### Problem 16:
- Points: \((-2, 4)\) and \((0, 6)\)

Calculate the slope:
\[
m = \frac{6 - 4}{0 - (-2)} = \frac{6 - 4}{2} = \frac{2}{2} = 1
\]

Use the point-slope form with \((-2, 4)\):
\[
y - 4 = 1(x - (-2))
\]
\[
y - 4 = 1(x + 2)
\]
\[
y - 4 = x + 2
\]
\[
y = x + 2 + 4
\]
\[
y = x + 6
\]

Answer:
\[
\boxed{y = x + 6}
\]

#### Problem 17:
- Points: \((3, 3)\) and \((1, -3)\)

Calculate the slope:
\[
m = \frac{-3 - 3}{1 - 3} = \frac{-3 - 3}{-2} = \frac{-6}{-2} = 3
\]

Use the point-slope form with \((3, 3)\):
\[
y - 3 = 3(x - 3)
\]
\[
y - 3 = 3x - 9
\]
\[
y = 3x - 9 + 3
\]
\[
y = 3x - 6
\]

Answer:
\[
\boxed{y = 3x - 6}
\]

#### Problem 18:
- Points: \((-1, 6)\) and \((3, -2)\)

Calculate the slope:
\[
m = \frac{-2 - 6}{3 - (-1)} = \frac{-2 - 6}{3 + 1} = \frac{-8}{4} = -2
\]

Use the point-slope form with \((-1, 6)\):
\[
y - 6 = -2(x - (-1))
\]
\[
y - 6 = -2(x + 1)
\]
\[
y - 6 = -2x - 2
\]
\[
y = -2x - 2 + 6
\]
\[
y = -2x + 4
\]

Answer:
\[
\boxed{y = -2x + 4}
\]

---

Section 3: Real-World Application

#### Problem 19:
The price of a share of stock in XYZ Corporation was \$74 two weeks ago and \$59 seven weeks ago.

##### Part (a): Write a linear equation to find the price \( p \) of a share of XYZ Corporation stock \( w \) weeks from now.

1. Identify the points:
- Two weeks ago (\( w = -2 \)): \( p = 74 \)
- Seven weeks ago (\( w = -7 \)): \( p = 59 \)

2. Calculate the slope \( m \):
\[
m = \frac{74 - 59}{-2 - (-7)} = \frac{74 - 59}{-2 + 7} = \frac{15}{5} = 3
\]

3. Use the point-slope form with \((-2, 74)\):
\[
p - 74 = 3(w - (-2))
\]
\[
p - 74 = 3(w + 2)
\]
\[
p - 74 = 3w + 6
\]
\[
p = 3w + 6 + 74
\]
\[
p = 3w + 80
\]

Answer:
\[
\boxed{p = 3w + 80}
\]

##### Part (b): Estimate the price of a share of stock five weeks ago.

1. Substitute \( w = -5 \) into the equation \( p = 3w + 80 \):
\[
p = 3(-5) + 80
\]
\[
p = -15 + 80
\]
\[
p = 65
\]

Answer:
\[
\boxed{65}
\]

---

Final Answers:

1. \(\boxed{y = -3x + 1}\)
2. \(\boxed{y = x - 3}\)
3. \(\boxed{y = 2x + 4}\)
4. \(\boxed{y = 4x + 5}\)
5. \(\boxed{y = -2x + 10}\)
6. \(\boxed{y = 3x - 8}\)
7. \(\boxed{y = 5x - 15}\)
8. \(\boxed{y = 2x + 4}\)
9. \(\boxed{y = -4x - 16}\)
10. \(\boxed{y = -x + 1}\)
11. \(\boxed{y = 2x - 1}\)
12. \(\boxed{y = -2x + 3}\)
13. \(\boxed{y = 2x + 1}\)
14. \(\boxed{y = -x + 5}\)
15. \(\boxed{y = 3x - 4}\)
16. \(\boxed{y = x + 6}\)
17. \(\boxed{y = 3x - 6}\)
18. \(\boxed{y = -2x + 4}\)
19. (a) \(\boxed{p = 3w + 80}\), (b) \(\boxed{65}\)