What is a Linear Equation with Fractions? | Simple Explanation for Class 6

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What is a Linear Equation with Fractions on Both Sides?

Grade Level:

Class 6

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

Definition

What is it?

A linear equation with fractions on both sides is a mathematical sentence where an unknown value (like 'x') is present, and both sides of the '=' sign have terms written as fractions. The highest power of the unknown variable is always 1.

Simple Example

Quick Example

Imagine you are sharing a pizza. If 1/3 of your pizza is equal to 1/2 of your friend's pizza, and your friend's pizza is 2 slices more than yours, how many slices does your pizza have? This kind of problem uses a linear equation with fractions on both sides to find the unknown number of slices.

Worked Example

Step-by-Step

Let's solve: (x/2) + 1 = (x/3) + 2

1. First, we want to bring all 'x' terms to one side and numbers to the other. Subtract x/3 from both sides: (x/2) - (x/3) + 1 = 2

2. Subtract 1 from both sides: (x/2) - (x/3) = 2 - 1

3. Simplify the right side: (x/2) - (x/3) = 1

4. Find the Least Common Multiple (LCM) of the denominators (2 and 3), which is 6. Convert the fractions to have a common denominator: (3x/6) - (2x/6) = 1

5. Combine the fractions on the left side: (3x - 2x)/6 = 1

6. Simplify the numerator: x/6 = 1

7. Multiply both sides by 6 to isolate x: x = 1 * 6

8. So, x = 6.

Answer: x = 6

Why It Matters

Understanding these equations is like learning a secret code that helps solve real-world puzzles! Engineers use them to design bridges, economists use them to predict market trends, and even computer scientists use them in programming. Mastering this helps you think logically, which is key for future careers in AI/ML or Data Science.

Common Mistakes

MISTAKE: Adding or subtracting fractions without a common denominator. For example, x/2 + x/3 = 2x/5 | CORRECTION: Always find the LCM of the denominators first to make them the same before adding or subtracting fractions. x/2 + x/3 = 3x/6 + 2x/6 = 5x/6

MISTAKE: Forgetting to apply an operation to BOTH sides of the equation. For example, if you multiply the left side by 6, but not the right side. | CORRECTION: Whatever operation you perform on one side of the '=' sign, you must perform the exact same operation on the other side to keep the equation balanced.

MISTAKE: Incorrectly simplifying terms, especially with negative signs. For example, 3x - 2x is simplified to just 1x, not 1. | CORRECTION: Pay close attention to the coefficients and signs when combining like terms. 3x - 2x = (3-2)x = 1x = x.

Practice Questions

Try It Yourself

QUESTION: Solve for y: (y/4) - 2 = (y/5) | ANSWER: y = 40

QUESTION: Find the value of z: (2z/3) + 5 = (z/2) + 8 | ANSWER: z = 18

QUESTION: A shopkeeper sells 1/4 of his samosas in the morning and 1/3 of the remaining samosas in the afternoon. If he has 20 samosas left, how many did he start with? (Hint: Let 'x' be the total samosas. Remaining after morning = x - x/4. Then take 1/3 of that.) | ANSWER: 40 samosas

MCQ

Quick Quiz

Which of these is the first step to solve (x/3) + 2 = (x/4) + 3?

Multiply both sides by 12

Subtract 2 from both sides

Subtract x/4 from both sides

Add 3 to both sides

The Correct Answer Is:

The goal is to gather all 'x' terms on one side. Subtracting x/4 from both sides is the most direct first step to achieve this. While multiplying by 12 is also a valid step, it's not typically the *first* step for isolating variables.

Real World Connection

In the Real World

Imagine you're planning a train journey from Mumbai to Delhi. If one train covers 1/3 of the distance in 't' hours, and another train covers 1/4 of the distance plus 50 km in the same 't' hours, an equation with fractions can help you find 't' or the total distance. This is how logistics companies like Delhivery or Ecom Express optimize delivery routes and timings.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1 | FRACTION: A part of a whole, written as numerator/denominator | VARIABLE: A letter (like x, y) representing an unknown value | DENOMINATOR: The bottom number in a fraction, showing how many equal parts make up the whole | LCM (Least Common Multiple): The smallest positive number that is a multiple of two or more numbers

What's Next

What to Learn Next

Great job with linear equations! Next, you can explore 'Linear Equations with Decimals' or 'Equations with Brackets'. These concepts build on what you've learned here, making problem-solving even more exciting and powerful!