What is a Linear Equation in Two Variables?

S6-SA1-0002

Grade Level:

Class 10

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

Definition

What is it?

A linear equation in two variables is an equation that involves two different unknown values, usually represented by 'x' and 'y'. The highest power of each variable in the equation is always 1. These equations represent a straight line when plotted on a graph.

Simple Example

Quick Example

Imagine you go to a chai stall. You buy some cups of chai (let's say 'x' cups) and some plates of samosa (let's say 'y' plates). If each chai costs Rs. 10 and each samosa costs Rs. 20, and your total bill was Rs. 100, we can write this as: 10x + 20y = 100. Here, 'x' and 'y' are the two variables.

Worked Example

Step-by-Step

Let's find if (2, 3) is a solution to the equation 3x + 2y = 12.
---1. In the given equation, 'x' represents the first value in the pair (2) and 'y' represents the second value (3).
---2. Substitute x = 2 into the equation: 3 * (2) = 6.
---3. Substitute y = 3 into the equation: 2 * (3) = 6.
---4. Now, add these values together: 6 + 6 = 12.
---5. Compare this result with the right side of the equation (12).
---6. Since 12 = 12, the values (2, 3) satisfy the equation.
---Answer: Yes, (2, 3) is a solution to the equation 3x + 2y = 12.

Why It Matters

Understanding linear equations helps in predicting outcomes and making decisions. Engineers use them to design structures, while data scientists use them in AI models to find relationships between different factors. Doctors can also use them to calculate medicine dosages based on a patient's weight and age.

Common Mistakes

MISTAKE: Confusing a linear equation with a quadratic equation (e.g., x^2 + y = 5). | CORRECTION: Remember that in a linear equation, the highest power of any variable is always 1 (like x or y, not x^2 or y^3).

MISTAKE: Forgetting to substitute both x and y values when checking a solution. | CORRECTION: Always replace both 'x' and 'y' with their given numerical values in the equation to see if the left side equals the right side.

MISTAKE: Incorrectly performing arithmetic operations (addition, subtraction, multiplication) after substitution. | CORRECTION: Double-check all calculations carefully, especially when dealing with negative numbers or fractions.

Practice Questions

Try It Yourself

QUESTION: Is (1, 4) a solution to the equation 5x + y = 9? | ANSWER: Yes

QUESTION: Find the value of 'k' if (2, -1) is a solution to the equation 3x - ky = 8. | ANSWER: k = -2

QUESTION: A farmer has chickens and goats. If there are a total of 20 heads and 56 legs, write two linear equations in two variables (x for chickens, y for goats) that represent this situation. | ANSWER: x + y = 20 and 2x + 4y = 56

MCQ

Quick Quiz

Which of the following is a linear equation in two variables?

x^2 + y = 7

3x + 2y = 10

5x - 4 = 0

xy = 5

The Correct Answer Is:

Option B (3x + 2y = 10) has two variables (x and y) and the highest power of each is 1. Options A and D have variables with powers higher than 1 or multiplied together, making them non-linear. Option C has only one variable.

Real World Connection

In the Real World

Think about planning a road trip across India. You might use a map app that calculates the best route based on average speed (x) and time taken (y) for different road sections, often using linear equations to optimize travel time and fuel consumption. Even delivery services like Swiggy or Zomato use these concepts to figure out the most efficient routes for their riders.

Key Vocabulary

Key Terms

VARIABLE: A letter representing an unknown value, like 'x' or 'y'. | COEFFICIENT: The number multiplied by a variable, like '3' in '3x'. | CONSTANT: A number without a variable, like '10' in '3x + 2y = 10'. | SOLUTION: A pair of values (x, y) that makes the equation true. | LINEAR: Describes an equation whose graph is a straight line.

What's Next

What to Learn Next

Great job learning about linear equations in two variables! Next, you should explore 'Pairs of Linear Equations in Two Variables'. This will teach you how to solve two such equations together to find a unique solution, which is super useful for real-world problems!