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What is the Distributive Property in Algebra?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Distributive Property in Algebra tells us how to multiply a single term by a group of terms inside parentheses. It means you 'distribute' or multiply the outside term with each term inside the parentheses separately, and then add or subtract the results.
Simple Example
Quick Example
Imagine you buy 2 packets of biscuits, and each packet has 5 chocolate biscuits and 3 plain biscuits. Instead of adding (5+3) first and then multiplying by 2, you can multiply 2 by 5 (chocolate) and 2 by 3 (plain) separately, then add the results. Both ways give you the total number of biscuits.
Worked Example
Step-by-Step
Let's solve 4 * (5 + 3) using the Distributive Property.
STEP 1: Identify the term outside the parentheses and the terms inside. Here, 4 is outside, and 5 and 3 are inside.
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STEP 2: Distribute (multiply) the outside term (4) with the first term inside (5).
4 * 5 = 20
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STEP 3: Distribute (multiply) the outside term (4) with the second term inside (3).
4 * 3 = 12
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STEP 4: Add the results from Step 2 and Step 3.
20 + 12 = 32
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So, 4 * (5 + 3) = 32. (You can check: 4 * 8 = 32)
Why It Matters
Understanding the Distributive Property is key to solving complex equations in higher math and science. It's used by engineers to design structures, by computer programmers to write efficient code, and even in economics to model financial situations. Mastering it helps you build a strong foundation for future careers in AI, Data Science, and Engineering.
Common Mistakes
MISTAKE: Multiplying the outside term by only the first term inside the parentheses. For example, in 3 * (x + 2), writing 3x + 2. | CORRECTION: Remember to multiply the outside term by EVERY term inside the parentheses. So, 3 * (x + 2) becomes 3x + 3*2, which is 3x + 6.
MISTAKE: Forgetting to apply the negative sign when distributing. For example, in 2 * (x - 3), writing 2x - 3. | CORRECTION: The sign with the term inside the parentheses also gets distributed. So, 2 * (x - 3) becomes 2x + 2*(-3), which is 2x - 6.
MISTAKE: Confusing distribution with simple multiplication of terms. For example, thinking 2x * (y + 3) is the same as 2x * y * 3. | CORRECTION: The distributive property applies when you have addition or subtraction inside the parentheses. You multiply the outside term by each term inside, then keep the operation (addition/subtraction) between them.
Practice Questions
Try It Yourself
QUESTION: Solve 5 * (x + 7) using the Distributive Property. | ANSWER: 5x + 35
QUESTION: Simplify 6 * (2y - 4). | ANSWER: 12y - 24
QUESTION: A vegetable vendor sells 3 kg of potatoes and 2 kg of onions. If the price of potatoes is Rs. 25/kg and onions is Rs. 30/kg, write an expression for the total cost using the Distributive Property, and then calculate the total cost. | ANSWER: Cost = (3 * 25) + (2 * 30) = 75 + 60 = Rs. 135 (Note: While not a direct distributive problem in the form a*(b+c), it's an application of distributing costs across items, which can be thought of as a distributive idea in real life.) A more direct example would be 5 * (price of potato + price of onion) if buying 5kg of a mix. Let's rephrase for clarity: Simplify 7 * (a - 8). | ANSWER: 7a - 56
MCQ
Quick Quiz
Which expression is equivalent to 8 * (p + 4)?
8p + 4
8p + 32
12p
p + 32
The Correct Answer Is:
B
The correct answer is B because you must multiply 8 by 'p' (giving 8p) AND multiply 8 by '4' (giving 32). Then you add these two results, making it 8p + 32. Options A, C, and D do not correctly apply the multiplication to both terms inside the parentheses.
Real World Connection
In the Real World
Imagine you are building a budget for a school trip. If 20 students are going, and each student needs money for a bus ticket (Rs. 50) and snacks (Rs. 30), you can use the Distributive Property. Total cost = 20 * (50 + 30) = (20 * 50) + (20 * 30). This helps event planners or small business owners calculate costs efficiently.
Key Vocabulary
Key Terms
DISTRIBUTE: To multiply an outside term by each term inside parentheses | PARENTHESES: The symbols ( ) used to group terms together | TERM: A single number or variable, or numbers and variables multiplied together | EXPRESSION: A mathematical phrase that can contain numbers, variables, and operators but no equals sign
What's Next
What to Learn Next
Great job understanding the Distributive Property! Next, you should explore 'Factoring Algebraic Expressions'. Factoring is like doing the Distributive Property in reverse, which is a very powerful tool for solving more complex equations and will help you master algebra.
