What is Elimination of One Variable? | Simple Explanation for Class 10

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What is the Elimination of One Variable?

Grade Level:

Class 10

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

Definition

What is it?

The elimination of one variable is a method used to solve a system of two linear equations with two variables. The main idea is to get rid of one variable by adding or subtracting the equations, so you are left with a single equation that has only one variable, which is easier to solve.

Simple Example

Quick Example

Imagine you bought 2 samosas and 1 chai for Rs. 50. Your friend bought 3 samosas and 1 chai for Rs. 70. If we want to find the price of just one samosa, we can 'eliminate' the chai variable. By subtracting your friend's bill from yours, the chai price cancels out, leaving only the samosa prices to compare.

Worked Example

Step-by-Step

Let's solve for x and y:
Equation 1: 2x + y = 10
Equation 2: 3x + y = 13

1. Notice that the 'y' terms in both equations have the same coefficient (1) and the same sign (+).
---2. To eliminate 'y', subtract Equation 1 from Equation 2:
(3x + y) - (2x + y) = 13 - 10
---3. Simplify the equation:
3x + y - 2x - y = 3
x = 3
---4. Now that we have the value of x, substitute x = 3 into either Equation 1 or Equation 2. Let's use Equation 1:
2(3) + y = 10
---5. Solve for y:
6 + y = 10
y = 10 - 6
y = 4
---6. So, the solution is x = 3 and y = 4.

Why It Matters

Elimination of variables is fundamental in solving complex problems across many fields. Engineers use it to design structures, scientists in Chemistry and Physics use it to model reactions or forces, and even AI/ML algorithms rely on similar principles to find optimal solutions. It's a core skill for future innovators!

Common Mistakes

MISTAKE: Not making the coefficients of the variable to be eliminated equal before adding or subtracting. | CORRECTION: Always multiply one or both equations by suitable numbers so that the coefficients of one variable become the same (or additive inverses) before performing addition or subtraction.

MISTAKE: Making sign errors when subtracting entire equations, especially with negative terms. | CORRECTION: When subtracting an entire equation, remember to change the sign of EVERY term in the equation being subtracted before combining.

MISTAKE: Forgetting to find the value of the second variable after finding the first. | CORRECTION: Once you find the value of one variable, always substitute it back into one of the original equations to find the value of the other variable.

Practice Questions

Try It Yourself

QUESTION: Solve using elimination: x + y = 7 and x - y = 3 | ANSWER: x = 5, y = 2

QUESTION: Solve using elimination: 3x + 2y = 13 and 3x - y = 4 | ANSWER: x = 3, y = 2

QUESTION: Solve using elimination: 2x + 3y = 11 and 5x - 2y = 8 | ANSWER: x = 2, y = 7/3 (or 2.33 approx)

MCQ

Quick Quiz

To eliminate 'y' from the equations 4x + 3y = 10 and 2x - y = 5, what is the first step you should take?

Add the two equations directly.

Subtract the second equation from the first.

Multiply the second equation by 3.

Multiply the first equation by 2.

The Correct Answer Is:

To eliminate 'y', its coefficients must be the same or opposite. Multiplying the second equation (2x - y = 5) by 3 makes it 6x - 3y = 15. Now the 'y' coefficients are +3y and -3y, which can be eliminated by adding the equations.

Real World Connection

In the Real World

Imagine you work for a company like Zomato or Swiggy and need to optimize delivery routes. You might have equations representing fuel cost and time for different routes. Using elimination, you can solve these equations to find the best route that minimizes both cost and time, ensuring faster and cheaper deliveries for customers across India.

Key Vocabulary

Key Terms

Variable: A letter representing an unknown value, like 'x' or 'y' | Coefficient: The number multiplied by a variable, e.g., '3' in 3x | Linear Equation: An equation where the highest power of the variable is 1 | System of Equations: A set of two or more equations with the same variables | Elimination Method: A technique to solve systems of equations by removing one variable

What's Next

What to Learn Next

Great job understanding elimination! Now that you can solve systems of equations, you're ready to explore the 'Substitution Method' for solving them. It's another powerful tool that builds on your understanding of variables and equations, giving you more ways to tackle problems.