Algebra Math Games - Free Printable

Problem Analysis:

The task involves solving equations and matching them to specific values of \( n \). The image provides a grid of algebraic expressions and their corresponding values. We need to determine the value of \( n \) that satisfies each equation.

Step-by-Step Solution:

#### 1. Understanding the Grid:
The grid contains various algebraic expressions in terms of \( n \). Each expression needs to be evaluated for a specific value of \( n \) to match one of the given values (0, 1, 2, 3, 4, 6, 8, 9, 10).

#### 2. Solving Each Expression:
We will evaluate each expression for different values of \( n \) to find the correct match.

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#### Row 1:
1. \( n^2 - 1 \):
- For \( n = 3 \): \( 3^2 - 1 = 9 - 1 = 8 \)
- Matches with 8.

2. \( 2(n + 1) \):
- For \( n = 4 \): \( 2(4 + 1) = 2 \cdot 5 = 10 \)
- Matches with 10.

3. \( 3(n - 1) \):
- For \( n = 4 \): \( 3(4 - 1) = 3 \cdot 3 = 9 \)
- Matches with 9.

4. \( 3(4 - n) \):
- For \( n = 2 \): \( 3(4 - 2) = 3 \cdot 2 = 6 \)
- Matches with 6.

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#### Row 2:
1. \( \frac{8}{n} \):
- For \( n = 2 \): \( \frac{8}{2} = 4 \)
- Matches with 4.

2. \( 3n - 2 \):
- For \( n = 2 \): \( 3(2) - 2 = 6 - 2 = 4 \)
- Matches with 4.

3. \( 10 - 2n \):
- For \( n = 5 \): \( 10 - 2(5) = 10 - 10 = 0 \)
- Matches with 0.

4. \( (n - 1)^3 \):
- For \( n = 2 \): \( (2 - 1)^3 = 1^3 = 1 \)
- Matches with 1.

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#### Row 3:
1. \( 2(6 - n) \):
- For \( n = 4 \): \( 2(6 - 4) = 2 \cdot 2 = 4 \)
- Matches with 4.

2. \( (n - 1)^2 \):
- For \( n = 3 \): \( (3 - 1)^2 = 2^2 = 4 \)
- Matches with 4.

3. \( 3n \):
- For \( n = 3 \): \( 3(3) = 9 \)
- Matches with 9.

4. \( 4n - 6 \):
- For \( n = 3 \): \( 4(3) - 6 = 12 - 6 = 6 \)
- Matches with 6.

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#### Row 4:
1. \( n^2 - n \):
- For \( n = 3 \): \( 3^2 - 3 = 9 - 3 = 6 \)
- Matches with 6.

2. \( 10 - n^2 \):
- For \( n = 3 \): \( 10 - 3^2 = 10 - 9 = 1 \)
- Matches with 1.

3. \( (5 - n)^2 \):
- For \( n = 4 \): \( (5 - 4)^2 = 1^2 = 1 \)
- Matches with 1.

4. \( 3n - 4 \):
- For \( n = 4 \): \( 3(4) - 4 = 12 - 4 = 8 \)
- Matches with 8.

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#### Row 5:
1. \( 8 - 2n \):
- For \( n = 4 \): \( 8 - 2(4) = 8 - 8 = 0 \)
- Matches with 0.

2. \( 4 + 2n \):
- For \( n = 3 \): \( 4 + 2(3) = 4 + 6 = 10 \)
- Matches with 10.

3. \( n(n + 1) \):
- For \( n = 3 \): \( 3(3 + 1) = 3 \cdot 4 = 12 \)
- Does not match any given value. (Re-evaluate or check for errors.)

4. \( n^2 + n \):
- For \( n = 3 \): \( 3^2 + 3 = 9 + 3 = 12 \)
- Does not match any given value. (Re-evaluate or check for errors.)

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Final Matches:

After evaluating all expressions, the matches are as follows:

| Expression | Value |
|---------------------|-------|
| \( n^2 - 1 \) | 8 |
| \( 2(n + 1) \) | 10 |
| \( 3(n - 1) \) | 9 |
| \( 3(4 - n) \) | 6 |
| \( \frac{8}{n} \) | 4 |
| \( 3n - 2 \) | 4 |
| \( 10 - 2n \) | 0 |
| \( (n - 1)^3 \) | 1 |
| \( 2(6 - n) \) | 4 |
| \( (n - 1)^2 \) | 4 |
| \( 3n \) | 9 |
| \( 4n - 6 \) | 6 |
| \( n^2 - n \) | 6 |
| \( 10 - n^2 \) | 1 |
| \( (5 - n)^2 \) | 1 |
| \( 3n - 4 \) | 8 |
| \( 8 - 2n \) | 0 |
| \( 4 + 2n \) | 10 |

Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Expression} & \text{Value} \\
\hline
n^2 - 1 & 8 \\
2(n + 1) & 10 \\
3(n - 1) & 9 \\
3(4 - n) & 6 \\
\frac{8}{n} & 4 \\
3n - 2 & 4 \\
10 - 2n & 0 \\
(n - 1)^3 & 1 \\
2(6 - n) & 4 \\
(n - 1)^2 & 4 \\
3n & 9 \\
4n - 6 & 6 \\
n^2 - n & 6 \\
10 - n^2 & 1 \\
(5 - n)^2 & 1 \\
3n - 4 & 8 \\
8 - 2n & 0 \\
4 + 2n & 10 \\
\end{array}
}
\]