Algebra 1 - Abeka 9th Grade 9 Highschool Mathematics Student Work Text - Free Printable

It looks like you've uploaded a solution key for Chapter 1: Basic Algebra, specifically Practice 1.1 from pages 7–8 of a textbook or workbook. The page includes:

- Solutions to 8 algebraic expressions (problems 1–8), showing step-by-step simplifications.
- Answers to 10 property identification questions (problems 9–18), identifying which algebraic property is used in each case.

However, since this is a solution key, it doesn't contain the original problems — only the solutions and explanations. You're likely asking me to solve and explain the problems based on what's shown here.

Let’s go through each problem, reconstruct the original expression or question, and explain how the solution was reached, step by step.

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🔹 Problems 1–8: Simplify the Expressions

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#### Problem 1:
Expression:
$$
\frac{8}{4} + (7 - 5) \cdot 2
$$

Solution Steps:
1. Simplify fractions and parentheses:
$$
\frac{8}{4} = 2,\quad (7 - 5) = 2
$$
2. Now:
$$
2 + 2 \cdot 2
$$
3. Multiplication before addition:
$$
2 + 4 = 6
$$

✔ Answer: 6

🔹 Explanation: Follow order of operations (PEMDAS): Parentheses → Exponents → Multiplication/Division → Addition/Subtraction.

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#### Problem 2:
Expression:
$$
\frac{6}{4} - (-10 - 3) \cdot 5
$$

Solution Steps:
1. Simplify inside parentheses:
$$
-10 - 3 = -13
$$
2. So:
$$
\frac{6}{4} - (-13) \cdot 5
$$
3. Simplify fraction:
$$
\frac{6}{4} = \frac{3}{2}
$$
4. Multiply:
$$
(-13) \cdot 5 = -65
$$
5. Now:
$$
\frac{3}{2} - (-65) = \frac{3}{2} + 65
$$
6. Convert 65 to fraction:
$$
65 = \frac{130}{2},\quad \frac{3}{2} + \frac{130}{2} = \frac{133}{2}
$$
7. As mixed number: $66\frac{1}{2}$

✔ Answer: $\frac{133}{2}$ or $66\frac{1}{2}$

🔹 Explanation: Negative times negative becomes positive when subtracting a negative.

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#### Problem 3:
Expression:
$$
8^2 - 6 + 5(1 + 2)
$$

Solution Steps:
1. Exponent:
$$
8^2 = 64
$$
2. Parentheses:
$$
1 + 2 = 3
$$
3. Multiply:
$$
5 \cdot 3 = 15
$$
4. Now:
$$
64 - 6 + 15
$$
5. Left to right:
$$
64 - 6 = 58,\quad 58 + 15 = 73
$$

✔ Answer: 73

🔹 Explanation: Order of operations: exponents first, then multiplication, then left-to-right addition/subtraction.

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#### Problem 4:
Expression:
$$
24 - 3(2^2 + 8) + 9 \div 3
$$

Solution Steps:
1. Exponent:
$$
2^2 = 4
$$
2. Inside parentheses:
$$
4 + 8 = 12
$$
3. Multiply:
$$
3 \cdot 12 = 36
$$
4. Division:
$$
9 \div 3 = 3
$$
5. Now:
$$
24 - 36 + 3
$$
6. Left to right:
$$
24 - 36 = -12,\quad -12 + 3 = -9
$$

✔ Answer: -9

🔹 Explanation: Important: Division happens before subtraction/addition, but all are at same level after that, so left to right.

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#### Problem 5:
Expression:
$$
3 + \frac{6 + 5 \cdot 3}{(8 - 1)} - 9
$$

Solution Steps:
1. Numerator: $6 + 5 \cdot 3 = 6 + 15 = 21$
2. Denominator: $8 - 1 = 7$
3. Fraction: $\frac{21}{7} = 3$
4. Now:
$$
3 + 3 - 9 = 6 - 9 = -3
$$

✔ Answer: -3

🔹 Explanation: Evaluate numerator and denominator separately, then simplify.

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#### Problem 6:
Expression:
$$
\frac{125 - 13 \cdot 2 + 34}{3 \cdot 4 - 5(24 + 5)}
$$

Solution Steps:
1. Numerator:
$$
13 \cdot 2 = 26,\quad 125 - 26 + 34 = 99 + 34 = 133
$$
2. Denominator:
$$
24 + 5 = 29,\quad 5 \cdot 29 = 145,\quad 3 \cdot 4 = 12,\quad 12 - 145 = -133
$$
3. Fraction:
$$
\frac{133}{-133} = -1
$$

✔ Answer: -1

🔹 Explanation: Careful with signs and order of operations in both numerator and denominator.

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#### Problem 7:
Expression:
$$
5^3 + \frac{8 - 4}{2}(15 - 2^2) - 12
$$

Solution Steps:
1. $5^3 = 125$
2. $8 - 4 = 4$, so $\frac{4}{2} = 2$
3. $2^2 = 4$, so $15 - 4 = 11$
4. Multiply: $2 \cdot 11 = 22$
5. Now:
$$
125 + 22 - 12 = 147 - 12 = 135
$$

✔ Answer: 135

🔹 Explanation: Fraction simplifies to 2, then multiply by result of parentheses.

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#### Problem 8:
Expression:
$$
\frac{3^2 + 27}{4 - 8} + 8 \cdot \frac{1}{2}
$$

Solution Steps:
1. $3^2 = 9$, so numerator: $9 + 27 = 36$
2. Denominator: $4 - 8 = -4$
3. Fraction: $\frac{36}{-4} = -9$
4. $8 \cdot \frac{1}{2} = 4$
5. Add: $-9 + 4 = -5$

✔ Answer: -5

🔹 Explanation: Simplify each term separately before combining.

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🔹 Problems 9–18: Identify Properties

These are about algebraic properties. Here's what each one means:

| Problem | Property | Explanation |
|--------|---------|-----------|
| 9. Associative property of multiplication | $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ | Grouping doesn't affect product. |
| 10. Associative property of addition | $(a + b) + c = a + (b + c)$ | Grouping doesn't affect sum. |
| 11. Inverse property of addition | $a + (-a) = 0$ | Adding opposite gives zero. |
| 12. Inverse property of multiplication | $a \cdot \frac{1}{a} = 1$ (for $a \neq 0$) | Multiplying by reciprocal gives 1. |
| 13. Commutative property of multiplication | $a \cdot b = b \cdot a$ | Order doesn’t matter. |
| 14. Commutative property of addition | $a + b = b + a$ | Order doesn’t matter. |
| 15. Identity property of addition | $a + 0 = a$ | Adding zero doesn’t change value. |
| 16. Identity property of multiplication | $a \cdot 1 = a$ | Multiplying by 1 doesn’t change value. |
| 17. Distributive property | $a(b + c) = ab + ac$ | Multiply over addition. |
| 18. Distributive property | Same as above — repeated. |

💡 These are foundational algebra concepts used to justify steps in solving equations.

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✔ Summary

This solution key provides correct answers and clear reasoning for basic algebra problems involving:

- Order of operations (PEMDAS)
- Simplifying expressions with fractions, exponents, parentheses
- Identifying fundamental algebraic properties

If you'd like, I can generate the original problems that match these solutions or create similar practice problems for review!

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