What is a Linear Equation with Decimals?

S3-SA1-0537

Grade Level:

Class 6

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

Definition

What is it?

A linear equation with decimals is an equation where the highest power of the variable is 1, and some numbers in the equation have decimal points. It's like a balance scale where both sides are equal, and you need to find the value of an unknown number, which might involve parts of whole numbers.

Simple Example

Quick Example

Imagine you bought 2 packets of biscuits and a small chai. Each biscuit packet costs Rs. 15.50. If your total bill was Rs. 38.00, how much did the chai cost? Here, if 'x' is the cost of chai, the equation would be 2 * 15.50 + x = 38.00.

Worked Example

Step-by-Step

Let's solve: 3x + 4.5 = 12.0

1. Our goal is to get 'x' by itself on one side of the equation.
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2. First, subtract 4.5 from both sides to remove the constant term from the left side:
3x + 4.5 - 4.5 = 12.0 - 4.5
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3. This simplifies to:
3x = 7.5
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4. Now, to get 'x' alone, divide both sides by 3:
3x / 3 = 7.5 / 3
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5. Perform the division:
x = 2.5

ANSWER: x = 2.5

Why It Matters

Understanding linear equations with decimals is crucial for many fields like Data Science, where you analyze data with precise measurements, or Physics, when calculating speeds or distances. Engineers use them to design structures, and economists use them to model financial situations, making these concepts essential for future innovators and problem-solvers.

Common Mistakes

MISTAKE: Not performing the same operation on both sides of the equation. Forgetting to subtract from both sides. | CORRECTION: Remember, an equation is a balance. Whatever you do to one side (add, subtract, multiply, divide), you MUST do to the other side to keep it balanced.

MISTAKE: Making errors in decimal addition, subtraction, multiplication, or division. Forgetting to align decimal points when adding/subtracting. | CORRECTION: Always double-check your decimal arithmetic. For addition/subtraction, line up the decimal points. For multiplication, count decimal places in the factors. For division, make the divisor a whole number.

MISTAKE: Confusing multiplication with addition when isolating the variable. Forgetting to divide when a number is multiplied by the variable. | CORRECTION: If a number is being added to 'x', subtract it. If a number is multiplying 'x', divide by it. Always use the inverse operation.

Practice Questions

Try It Yourself

QUESTION: Solve for y: y + 2.7 = 8.1 | ANSWER: y = 5.4

QUESTION: Solve for z: 4z = 10.8 | ANSWER: z = 2.7

QUESTION: A tailor bought 3 meters of cloth at Rs. 75.50 per meter. He also spent Rs. 15.50 on thread. If his total bill was Rs. 242.00, how much did the cloth cost per meter if 'x' is the cost per meter? (Hint: Formulate the equation first) | ANSWER: x = 75.50 (The question already gives the cost per meter, it's testing if you can set up the equation 3x + 15.50 = 242.00, and solve for x if the 75.50 wasn't given. If we assume 'x' is an unknown cost per meter: 3x + 15.50 = 242.00 => 3x = 226.50 => x = 75.50)

MCQ

Quick Quiz

What is the value of 'a' in the equation: 5a - 3.2 = 11.8?

3.2

The Correct Answer Is:

To solve 5a - 3.2 = 11.8, first add 3.2 to both sides: 5a = 11.8 + 3.2, which gives 5a = 15.0. Then, divide both sides by 5: a = 15.0 / 5, so a = 3.0.

Real World Connection

In the Real World

When you track your monthly mobile data usage, say you have 10 GB (10.0 GB) and use 0.75 GB each day, you can use linear equations with decimals to figure out how many days 'd' your data will last (0.75d = 10.0). Similarly, a shopkeeper calculates total cost when items have decimal prices, or a chef adjusts recipes using precise measurements like 0.5 cups of flour.

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an equals sign. | VARIABLE: A symbol (like x, y, a) representing an unknown value. | DECIMAL: A number that includes a decimal point, representing parts of a whole number. | LINEAR: Means the variable's highest power is 1 (no x^2 or x^3). | INVERSE OPERATION: An operation that undoes another (e.g., addition undoes subtraction).

What's Next

What to Learn Next

Great job mastering linear equations with decimals! Next, you can explore linear equations with fractions or even more complex multi-step linear equations. These will build on your current skills and prepare you for solving even bigger real-world problems.