Answer key for slope-intercept form worksheet from tables.

The task involves writing equations in slope-intercept form (\( y = mx + b \)) based on given tables of \( x \) and \( y \) values. Here's a detailed explanation of how to solve each problem:

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General Steps:

1. Find the Slope (\( m \)):
- Use the formula for slope:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
- Choose any two points \((x_1, y_1)\) and \((x_2, y_2)\) from the table.

2. Find the Y-Intercept (\( b \)):
- The y-intercept is the value of \( y \) when \( x = 0 \).
- If \( x = 0 \) is not given in the table, use the slope and any point from the table in the equation \( y = mx + b \) to solve for \( b \).

3. Write the Equation:
- Substitute the values of \( m \) and \( b \) into the slope-intercept form:
\[
y = mx + b
\]

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Solutions for Each Problem:

#### Problem 1:
| \( x \) | \( y \) |
|---------|---------|
| 0 | 6 |
| 1 | 9 |
| 2 | 12 |
| 3 | 15 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 6}{1 - 0} = \frac{3}{1} = 3
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = 6 \). So, \( b = 6 \).

- Equation:
\[
y = 3x + 6
\]

#### Problem 2:
| \( x \) | \( y \) |
|---------|---------|
| 0 | -5 |
| 1 | -6 |
| 2 | -7 |
| 3 | -8 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - (-5)}{1 - 0} = \frac{-6 + 5}{1} = \frac{-1}{1} = -1
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = -5 \). So, \( b = -5 \).

- Equation:
\[
y = -x - 5
\]

#### Problem 3:
| \( x \) | \( y \) |
|---------|---------|
| -1 | 4 |
| 0 | 1 |
| 1 | -2 |
| 2 | -5 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 4}{0 - (-1)} = \frac{1 - 4}{0 + 1} = \frac{-3}{1} = -3
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = 1 \). So, \( b = 1 \).

- Equation:
\[
y = -3x + 1
\]

#### Problem 4:
| \( x \) | \( y \) |
|---------|---------|
| 0 | 9 |
| 2 | 19 |
| 4 | 29 |
| 6 | 39 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19 - 9}{2 - 0} = \frac{10}{2} = 5
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = 9 \). So, \( b = 9 \).

- Equation:
\[
y = 5x + 9
\]

#### Problem 5:
| \( x \) | \( y \) |
|---------|---------|
| -6 | -8 |
| -3 | -3 |
| 0 | 2 |
| 3 | 7 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-8)}{-3 - (-6)} = \frac{-3 + 8}{-3 + 6} = \frac{5}{3}
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = 2 \). So, \( b = 2 \).

- Equation:
\[
y = \frac{5}{3}x + 2
\]

#### Problem 6:
| \( x \) | \( y \) |
|---------|---------|
| -6 | 15 |
| -4 | 14 |
| -2 | 13 |
| 0 | 12 |

- Slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 15}{-4 - (-6)} = \frac{14 - 15}{-4 + 6} = \frac{-1}{2}
\]

- Y-Intercept (\( b \)):
When \( x = 0 \), \( y = 12 \). So, \( b = 12 \).

- Equation:
\[
y = -\frac{1}{2}x + 12
\]

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Final Answers:

1. \( \text{slope: } 3, \text{ y-intercept: } 6, \text{ equation: } y = 3x + 6 \)
2. \( \text{slope: } -1, \text{ y-intercept: } -5, \text{ equation: } y = -x - 5 \)
3. \( \text{slope: } -3, \text{ y-intercept: } 1, \text{ equation: } y = -3x + 1 \)
4. \( \text{slope: } 5, \text{ y-intercept: } 9, \text{ equation: } y = 5x + 9 \)
5. \( \text{slope: } \frac{5}{3}, \text{ y-intercept: } 2, \text{ equation: } y = \frac{5}{3}x + 2 \)
6. \( \text{slope: } -\frac{1}{2}, \text{ y-intercept: } 12, \text{ equation: } y = -\frac{1}{2}x + 12 \)

\boxed{
\begin{aligned}
1. & \quad y = 3x + 6 \\
2. & \quad y = -x - 5 \\
3. & \quad y = -3x + 1 \\
4. & \quad y = 5x + 9 \\
5. & \quad y = \frac{5}{3}x + 2 \\
6. & \quad y = -\frac{1}{2}x + 12
\end{aligned}
}