What is a Linear Equation with Brackets?

S3-SA1-0274

Grade Level:

Class 6

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

Definition

What is it?

A linear equation with brackets is an equation where the highest power of the variable is 1, and it contains parentheses or 'brackets' that group terms together. To solve it, you first remove the brackets by multiplying the number outside with each term inside, and then solve the simpler linear equation.

Simple Example

Quick Example

Imagine you bought 2 packets of laddoos for your friends. Each packet has (3 laddoos + 1 extra laddoo for you). If the total laddoos are 8, we can write this as 2(x + 1) = 8, where 'x' is the number of laddoos for friends in each packet. Here, (x + 1) is inside brackets.

Worked Example

Step-by-Step

Let's solve: 3(x + 2) = 15

Step 1: Distribute the number outside the bracket. Multiply 3 by 'x' and 3 by '2'.
3 * x + 3 * 2 = 15
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Step 2: Simplify the multiplication.
3x + 6 = 15
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Step 3: Isolate the term with 'x'. Subtract 6 from both sides of the equation.
3x + 6 - 6 = 15 - 6
3x = 9
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Step 4: Isolate 'x'. Divide both sides by 3.
3x / 3 = 9 / 3
x = 3
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Answer: x = 3

Why It Matters

Understanding linear equations with brackets is crucial for solving more complex problems in science and technology. Engineers use them to design structures, and data scientists use them to build models. This skill is a stepping stone for careers in AI/ML, Physics, and Computer Science.

Common Mistakes

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

MISTAKE: Forgetting to change the sign when distributing a negative number. For example, in -2(x - 3), writing -2x - 6. | CORRECTION: Remember that a negative multiplied by a negative gives a positive. So, -2(x - 3) becomes -2x + 6.

MISTAKE: Adding or subtracting numbers from inside the bracket before distributing. For example, in 3(x + 2) = 15, adding x + 2 first. | CORRECTION: Always distribute the number outside the bracket first to remove the brackets, following the order of operations (BODMAS/PEMDAS).

Practice Questions

Try It Yourself

QUESTION: Solve: 5(y - 1) = 20 | ANSWER: y = 5

QUESTION: Solve: 2(3a + 4) = 22 | ANSWER: a = 7/3 or 2.33 (approx)

QUESTION: Solve: 4(x + 3) - 5 = 15 | ANSWER: x = 2

MCQ

Quick Quiz

Which step is correct for simplifying 3(2x - 5) = 9?

6x - 5 = 9

6x - 15 = 9

2x - 15 = 9

3(2x) - 5 = 9

The Correct Answer Is:

Option B is correct because the 3 outside the bracket must be multiplied by both 2x and -5. This gives 6x - 15. The other options show incorrect distribution.

Real World Connection

In the Real World

Imagine you are planning a cricket match and need to calculate the total runs for two teams. If Team A scored 'x' runs and Team B scored 10 more than twice Team A's score, you might write Team B's score as 2(x + 10). Linear equations with brackets help solve such real-life puzzles, like calculating costs for a family trip or figuring out how many samosas to order for a party.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1, like x + 5 = 10. | BRACKETS: Symbols like ( ) that group terms together in an expression. | DISTRIBUTE: To multiply a number outside a bracket by each term inside the bracket. | VARIABLE: A letter (like x, y, a) that represents an unknown number. | CONSTANT: A fixed numerical value in an equation, like 5 or 10.

What's Next

What to Learn Next

Now that you've mastered linear equations with brackets, you're ready to tackle more complex linear equations, including those with variables on both sides. This will further strengthen your problem-solving skills and prepare you for algebraic inequalities!