What is a Linear Equation with Parentheses?

S3-SA1-0535

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A linear equation with parentheses is an equation where the highest power of the variable (like 'x' or 'y') is 1, and it includes parts enclosed in round brackets (parentheses). To solve it, we first need to get rid of these brackets by using the distributive property.

Simple Example

Quick Example

Imagine your mother gives you some money for snacks. You buy 2 packets of biscuits, and each packet costs (₹10 + ₹5 for a small chocolate). So, the total cost would be 2 * (10 + 5). If the total bill was ₹30, the equation would be 2(10 + 5) = 30. Here, 2(x + 5) = 30 would be a linear equation with parentheses.

Worked Example

Step-by-Step

Solve for x: 3(x + 4) = 21

1. Distribute the number outside the parentheses to each term inside: 3 * x + 3 * 4 = 21
---2. Simplify the multiplication: 3x + 12 = 21
---3. Subtract 12 from both sides of the equation to isolate the term with 'x': 3x + 12 - 12 = 21 - 12
---4. Simplify: 3x = 9
---5. Divide both sides by 3 to find the value of 'x': 3x / 3 = 9 / 3
---6. Simplify: x = 3

Answer: x = 3

Why It Matters

Understanding linear equations with parentheses is crucial for problem-solving in science and technology. Engineers use them to design structures, and economists use them to model market trends. It's a foundational skill for careers in fields like AI, data science, and even space research at ISRO!

Common Mistakes

MISTAKE: Multiplying the number outside the parentheses with only the first term inside. For example, in 2(x + 3), writing 2x + 3. | CORRECTION: Multiply the number outside with EVERY term inside the parentheses. So, 2(x + 3) becomes 2x + 2*3, which is 2x + 6.

MISTAKE: Forgetting to distribute a negative sign outside the parentheses. For example, in -(x - 5), writing -x - 5. | CORRECTION: Remember that a negative sign outside means multiplying by -1. So, -(x - 5) becomes -1*x -1*(-5), which is -x + 5.

MISTAKE: Adding or subtracting numbers from inside the parentheses before distributing. For example, in 3(x + 2) = 15, changing it to 3(2x) = 15. | CORRECTION: Always apply the distributive property first, unless the terms inside the parentheses are like terms that can be combined. In 3(x + 2), 'x' and '2' are not like terms, so distribute first: 3x + 6 = 15.

Practice Questions

Try It Yourself

QUESTION: Solve for y: 5(y - 2) = 15 | ANSWER: y = 5

QUESTION: Solve for p: 2(p + 3) + 4 = 20 | ANSWER: p = 5

QUESTION: Solve for m: 4(m - 1) - 2m = 10 | ANSWER: m = 7

MCQ

Quick Quiz

Which of the following is the correct first step to solve the equation 4(x + 5) = 32?

Add 5 to both sides

Divide both sides by 4

Multiply 4 by x and 4 by 5

Subtract 5 from both sides

The Correct Answer Is:

The correct first step is to use the distributive property to remove the parentheses, which means multiplying 4 by both 'x' and '5'. Options A, B, and D are incorrect ways to start solving an equation with parentheses.

Real World Connection

In the Real World

Imagine you are calculating the total cost for a family trip. If you pay for 3 train tickets, and each ticket costs (₹500 + ₹50 for booking charges), the total cost would be 3(500 + 50). Linear equations with parentheses help you quickly figure out these kinds of total expenses, like when managing your pocket money or planning a small business budget.

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an equals sign (=). | PARENTHESES: Round brackets () used to group terms in an expression. | DISTRIBUTIVE PROPERTY: A rule that says a number multiplied by a sum is equal to the sum of the products (e.g., a(b+c) = ab + ac). | VARIABLE: A symbol (like x, y) that represents an unknown value in an equation. | TERM: A single number or variable, or numbers and variables multiplied together, separated by + or - signs.

What's Next

What to Learn Next

Great job learning about linear equations with parentheses! Next, you can explore 'Linear Equations with Variables on Both Sides'. This will build on your current skills and help you solve even more complex problems, preparing you for higher-level algebra.