Complete solutions for Algebra 1 variables and expressions practice problems from Chapter 1, Lesson 1-1 of the Glencoe curriculum.

You've uploaded a page from a Glencoe Algebra 1 workbook (Lesson 1-1: Variables and Expressions), showing both the Practice worksheet and its corresponding Answer Key.

Since you asked to “solve the problem and explain the solution,” I’ll walk through the key sections of this worksheet, explaining how to approach each type of problem — even though the answers are already provided. This will help you understand *how* to solve similar problems on your own.

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## 📘 SECTION 1: Write an algebraic expression for each verbal expression.

Goal:

Translate English phrases into mathematical expressions using variables.

Examples & Explanations:

#### 1. The difference of 10 and u
→ “Difference” means subtraction.
→ Answer: `10 - u`

#### 2. The sum of 18 and a number
→ “Sum” means addition. Let the number be `x`.
→ Answer: `18 + x`

#### 3. The product of 33 and j
→ “Product” means multiplication.
→ Answer: `33j` (or `33 × j`)

#### 4. 74 increased by 3 times y
→ “Increased by” = add; “3 times y” = `3y`
→ Answer: `74 + 3y`

#### 5. 15 decreased by twice a number
→ “Decreased by” = subtract; “twice a number” = `2x`
→ Answer: `15 - 2x`

#### 6. 91 more than the square of a number
→ “More than” = add; “square of a number” = `x²`
→ Answer: `x² + 91`

#### 7. Three fourths the square of b
→ “Three fourths” = `3/4`; “square of b” = `b²`
→ Answer: `(3/4)b²`

#### 8. Two fifths the cube of a number
→ “Two fifths” = `2/5`; “cube of a number” = `x³`
→ Answer: `(2/5)x³`

✔ Tip: Always identify the operation (add, subtract, multiply, divide) and the variable(s). Remember: “of” often means multiply, “more than” or “increased by” means add, etc.

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## 📐 SECTION 2: Evaluate each expression.

Goal:

Substitute the given value and compute the result.

Examples:

#### 9. `11²` → 11 × 11 = 121

#### 10. `8³` → 8 × 8 × 8 = 512

#### 11. `5⁴` → 5 × 5 × 5 × 5 = 625

#### 12. `4⁵` → 4 × 4 × 4 × 4 × 4 = 1024

#### 13. `9³` → 9 × 9 × 9 = 729

#### 14. `6⁴` → 6 × 6 × 6 × 6 = 1296

#### 15. `10⁵` → 10 × 10 × 10 × 10 × 10 = 100,000

#### 16. `12³` → 12 × 12 × 12 = 1728

#### 17. `10⁶` → 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000

✔ Tip: Know your powers! Practice squares and cubes up to 12. Also, remember that `10ⁿ` is just 1 followed by n zeros.

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## 🗣️ SECTION 3: Write a verbal expression for each algebraic expression.

Goal:

Reverse translation — turn math into words.

Examples:

#### 19. `23f`
→ “The product of 23 and f”

#### 20. `5²`
→ “5 squared”

#### 21. `4n³ - 10`
→ “Four times n cubed minus 10”

#### 22. `5m² + 2`
→ “Two more than five times m squared”

#### 23. `b² - 3c³`
→ “b squared minus three times c cubed”

#### 24. `k⁵ / 6`
→ “One sixth of the fifth power of k”

#### 25. `4t² / 7`
→ “One seventh of four times t squared”

✔ Tip: Read the expression left-to-right, naming operations as you go. Use phrases like “the product of,” “the square of,” “increased by,” etc.

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## 📚 SECTION 4: Word Problems

26. BOOKS — Used bookstore sells paperback fiction books:

- Excellent condition: $2.50
- Fair condition: $0.50

Write an expression for buying `e` excellent-condition paperbacks and `f` fair-condition paperbacks.

→ Cost = (price per excellent book × number) + (price per fair book × number)
→ Expression: `2.50e + 0.50f`

✔ Explanation: Multiply quantity by price for each category, then add.

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27. GEOMETRY — Surface area of side of a right cylinder.

Given: Multiply twice π by radius r and height h.

→ Formula: `2πrh`

✔ Explanation: The lateral surface area (side only, not top/bottom) of a cylinder is `2πr × h`, which is circumference (`2πr`) times height (`h`). So expression is `2πrh`.

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## ✔ Final Notes:

This worksheet reinforces foundational algebra skills:
- Translating between words and math.
- Evaluating exponents.
- Writing expressions for real-world situations.

Even though the answers are provided, understanding *why* each answer is correct is what builds true mastery.

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## 🧠 Need Help with Similar Problems?

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