What is a Linear Equation in One Variable?

S6-SA1-0001

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A linear equation in one variable is a mathematical statement where two expressions are equal, and it contains only one unknown quantity (variable) raised to the power of 1. Think of it like a balance scale where both sides must weigh the same, and you're trying to find the weight of one mystery item.

Simple Example

Quick Example

Imagine you bought a packet of biscuits for Rs 10 and a chai for some unknown price (let's call it 'x'). If your total bill was Rs 25, then the equation would be 10 + x = 25. Here, 'x' is the one variable, and it's a linear equation.

Worked Example

Step-by-Step

Let's solve for 'x' in the equation: 3x + 5 = 20
---Step 1: Our goal is to get 'x' by itself on one side. First, subtract 5 from both sides of the equation to isolate the term with 'x'.
3x + 5 - 5 = 20 - 5
---Step 2: Simplify both sides.
3x = 15
---Step 3: Now, 'x' is being multiplied by 3. To get 'x' alone, divide both sides by 3.
3x / 3 = 15 / 3
---Step 4: Simplify to find the value of 'x'.
x = 5
---Answer: The value of x is 5.

Why It Matters

Understanding linear equations is super important! Engineers use them to design bridges and calculate forces, while data scientists in AI/ML use them to build predictive models for things like weather forecasting or recommending movies. Even doctors use them to calculate medicine dosages for patients.

Common Mistakes

MISTAKE: Not performing the same operation on both sides of the equation. For example, adding 2 to the left side but not the right. | CORRECTION: Always remember to apply the exact same operation (addition, subtraction, multiplication, division) to both sides of the equation to maintain balance.

MISTAKE: Incorrectly combining terms, like adding 3x and 5 to get 8x. | CORRECTION: You can only add or subtract 'like terms'. You cannot combine a variable term (like 3x) with a constant term (like 5).

MISTAKE: Forgetting to change the sign of a term when moving it from one side of the equation to the other. For example, moving +7 to the other side as +7 instead of -7. | CORRECTION: When you move a term across the equals sign, always change its operation to the inverse (plus becomes minus, multiply becomes divide, and vice-versa).

Practice Questions

Try It Yourself

QUESTION: Solve for 'y': y - 7 = 15 | ANSWER: y = 22

QUESTION: If 4 times a number 'p' is 36, what is 'p'? Write an equation and solve it. | ANSWER: 4p = 36; p = 9

QUESTION: The total cost of 2 samosas and a cold drink is Rs 50. If the cold drink costs Rs 20, what is the price of one samosa (let 's' be the price of one samosa)? | ANSWER: 2s + 20 = 50; 2s = 30; s = 15

MCQ

Quick Quiz

Which of the following is a linear equation in one variable?

x^2 + 3 = 7

2x + 3y = 10

5x - 8 = 12

sqrt(x) + 4 = 9

The Correct Answer Is:

Option C (5x - 8 = 12) is a linear equation in one variable because it has only one variable ('x') and its highest power is 1. Options A and D have variables with powers other than 1, and Option B has two variables.

Real World Connection

In the Real World

When you buy groceries using a digital payment app like PhonePe or Google Pay, the app calculates your total bill. If you're buying 'x' number of items at a fixed price each, plus a delivery charge, the app uses a linear equation to quickly tell you the final amount. Even delivery services like Swiggy or Zomato use these to calculate the total cost of your order based on item prices and delivery fees.

Key Vocabulary

Key Terms

VARIABLE: A symbol (usually a letter like x, y, a) that represents an unknown number | CONSTANT: A fixed numerical value that does not change | EXPRESSION: A combination of numbers, variables, and operation symbols (e.g., 3x + 5) | EQUATION: A mathematical statement showing that two expressions are equal (e.g., 3x + 5 = 20) | LINEAR: Means the highest power of the variable is 1, so when graphed, it forms a straight line.

What's Next

What to Learn Next

Great job understanding linear equations in one variable! Next, you can explore 'Linear Equations in Two Variables'. This will help you solve problems with two unknowns, which is a step closer to real-world situations like figuring out the cost of two different items you buy.