Pre Algebra Week 2 Day 4 Answer Key

Pre-Algebra Week 2 Day 4 Answer Key: Mastering the Distributive Property

Welcome to the detailed breakdown of your Pre-Algebra Week 2 Day 4 assignment. In real terms, this day's focus is on one of the most fundamental and powerful tools in algebra: the distributive property. Understanding this concept is not just about completing a worksheet; it's about building a solid foundation that will support your success in all future math courses. Think of it as the secret key that simplifies complex problems, making them manageable and even enjoyable. In this thorough look, we will walk through each problem from the answer key, explaining not just what the answer is, but why it is the answer, ensuring you truly master the concept That's the part that actually makes a difference..

Introduction: What is the Distributive Property?

Before diving into the problems, let's solidify our understanding of the core concept. The distributive property of multiplication over addition (and subtraction) is a rule that allows us to multiply a single term by two or more terms inside a parenthesis. Its standard form is:

a × (b + c) = (a × b) + (a × c)

This might look simple, but it's a big shift. Instead of adding b and c first and then multiplying by a (which can be difficult if b and c are variables or large numbers), you can "distribute" the multiplication of a to each term inside the parenthesis separately. On top of that, this process is often called "expanding" an expression. The property works in reverse as well, which we call "factoring" or "undistributing," a skill we will also practice Easy to understand, harder to ignore. Worth knowing..

Problem Set 1: Applying the Distributive Property to Expand Expressions

The first set of problems is designed to get you comfortable with the forward application of the property. Your goal is to rewrite each expression without parentheses by distributing the multiplication.

1. 3(x + 4)

• Answer: 3x + 12
• Explanation: Here, the number 3 is outside the parentheses. We distribute this 3 to both the x and the 4 inside. This means we multiply 3 by x to get 3x, and we multiply 3 by 4 to get 12. The final expression is the sum of these two results: 3x + 12. It's crucial to remember that the sign between the terms inside the parentheses remains the same.

2. -5(y - 7)

• Answer: -5y + 35
• Explanation: Pay close attention to the negative sign! The number being distributed is -5. We multiply -5 by y to get -5y. Then, we multiply -5 by -7. Remember, a negative times a negative is a positive, so -5 × -7 = +35. The final answer is -5y + 35. A common mistake here is to write -5y - 35, forgetting that subtracting a negative is the same as adding a positive.

3. 2(3a + 2b - 6)

• Answer: 6a + 4b - 12
• Explanation: The distributive property works perfectly with more than two terms inside the parentheses. You simply distribute the 2 to each term. Multiply 2 by 3a to get 6a, 2 by 2b to get 4b, and 2 by -6 to get -12. The expression becomes 6a + 4b - 12.

4. (1/2)(8m + 10n)

• Answer: 4m + 5n
• Explanation: Fractions are no problem! Distribute the 1/2 to both terms. (1/2) × 8m = 4m and (1/2) × 10n = 5n. This demonstrates how the distributive property can simplify calculations by making numbers smaller and easier to work with.

Problem Set 2: Using the Distributive Property for Mental Math

This section tests your ability to use the property as a practical tool for calculation, just as you would in real-life situations.

5. Calculate 7 × 29 using the distributive property.

• Answer: 203
• Explanation: Instead of multiplying 7 × 29 directly, you can break 29 down into friendlier numbers, like 30 - 1. The problem then becomes 7 × (30 - 1). Now, distribute the 7: (7 × 30) - (7 × 1) = 210 - 7 = 203. This method is often much faster and less prone to error than traditional multiplication.

6. Calculate 12 × 102.

• Answer: 1,224
• Explanation: Similarly, break 102 into 100 + 2. The problem is 12 × (100 + 2). Distribute the 12: (12 × 100) + (12 × 2) = 1200 + 24 = 1,224. This strategy is incredibly useful for multiplying by numbers just above a multiple of 10.

Problem Set 3: Factoring (Reverse Distributive Property)

Factoring is the inverse operation of distributing. Here, you are given an expression like ax + ay and you need to find a common factor to "pull out" of each term, rewriting it as a(x + y). This skill is essential for solving equations and simplifying more complex algebraic expressions Small thing, real impact..

7. Factor: 15x + 45

• Answer: 15(x + 3)
• Explanation: Look for the Greatest Common Factor (GCF) of the two terms, 15x and 45. The factors of 15x are 1, 3, 5, 15, x, 3x, 5x, 15x. The factors of 45 are 1, 3, 5, 9, 15, 45. The GCF is 15. Now, divide each term by 15: 15x ÷ 15 = x and 45 ÷ 15 = 3. The factored expression is 15 multiplied by the sum of the quotients: 15(x + 3).

8. Factor: 8k - 12

• Answer: 4(2k - 3)
• Explanation: Find the GCF of 8k and 12. The factors of 8k include 1, 2, 4, 8, k, 2k, 4k, 8k. The factors of 12 are 1, 2, 3, 4, 6, 12. The GCF is 4. Divide each term by 4:

8k ÷ 4 = 2k and -12 ÷ 4 = -3. Write the GCF outside the parentheses and the quotients inside: 4(2k - 3). **Always check your work by distributing the 4 back in: 4(2k) + 4(-3) = 8k - 12 Less friction, more output..

9. Factor: -6x - 18

• Answer: -6(x + 3)
• Explanation: When the leading term is negative, it is standard practice to factor out a negative GCF. The GCF of 6x and 18 is 6, so we factor out -6. Divide each term by -6: -6x ÷ (-6) = x and -18 ÷ (-6) = 3. The result is -6(x + 3). Notice that the signs inside the parentheses flip compared to the original expression. Distributing -6 back through (x + 3) returns the original -6x - 18.

10. Factor: 12ab + 18a²

• Answer: 6a(2b + 3a)
• Explanation: This involves variables with exponents. Find the GCF of the coefficients (12 and 18 → 6) and the variables (a is common to both; b is only in the first term). The GCF is 6a. Divide each term by 6a: 12ab ÷ 6a = 2b and 18a² ÷ 6a = 3a. The factored form is 6a(2b + 3a).

Problem Set 4: Combining Like Terms After Distributing

Often, the distributive property is just the first step. Still, you must distribute first, then combine like terms to fully simplify the expression. Remember the order of operations: multiplication (distributing) comes before addition/subtraction (combining).

11. Simplify: 3(x + 4) + 2x

• Answer: 5x + 12
• Explanation: Distribute the 3 first: 3x + 12 + 2x. Now, identify and combine like terms (3x and 2x): 3x + 2x = 5x. The constant 12 remains. Final answer: 5x + 12.

12. Simplify: 5(2y - 3) - 4(y + 1)

• Answer: 6y - 19
• Explanation: Distribute the 5 and the -4 (watch the signs!): 5(2y) + 5(-3) - 4(y) - 4(1) = 10y - 15 - 4y - 4 Combine like terms: 10y - 4y = 6y and -15 - 4 = -19. Final answer: 6y - 19.

Conclusion

The distributive property is far more than a rule to memorize for a test—it is a fundamental lens through which to view the structure of mathematics. Whether you are breaking down 7 × 29 to calculate a tip mentally, expanding 3(x + 4) to solve for x, or factoring 6a(2b + 3a) to find the roots of a quadratic equation, you are relying on the exact same logic: multiplication distributes over addition and subtraction.

Mastery of this concept creates a bridge between arithmetic fluency and algebraic abstraction. By practicing both the forward motion (distributing/expanding) and the reverse motion (factoring), you build the bidirectional flexibility required for higher-level math. Keep an eye out for opportunities to use these strategies in daily life; the more you treat numbers as flexible, decomposable structures rather than rigid symbols, the more intuitive mathematics becomes.