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What is the Elimination Method for Linear Equations?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The Elimination Method is a technique used to solve a system of two linear equations with two variables. The main idea is to 'eliminate' one of the variables by adding or subtracting the equations, leaving you with a single equation that has only one variable, which is then easy to solve.
Simple Example
Quick Example
Imagine you went to a chai shop. You bought 2 cups of chai and 1 samosa for Rs 50. Your friend bought 1 cup of chai and 1 samosa for Rs 30. If we subtract your friend's bill from yours, the cost of the samosa gets 'eliminated', and you can easily find the cost of one cup of chai.
Worked Example
Step-by-Step
Let's solve the system of equations:
Equation 1: 2x + 3y = 10
Equation 2: x - 3y = 2
Step 1: Notice that the coefficients of 'y' are +3 and -3. They are already opposites, so if we add the equations, 'y' will be eliminated.
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Step 2: Add Equation 1 and Equation 2:
(2x + 3y) + (x - 3y) = 10 + 2
2x + x + 3y - 3y = 12
3x + 0y = 12
3x = 12
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Step 3: Solve for 'x':
x = 12 / 3
x = 4
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Step 4: Substitute the value of 'x' (which is 4) into either Equation 1 or Equation 2 to find 'y'. Let's use Equation 2:
x - 3y = 2
4 - 3y = 2
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Step 5: Solve for 'y':
-3y = 2 - 4
-3y = -2
y = -2 / -3
y = 2/3
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Answer: The solution is x = 4 and y = 2/3.
Why It Matters
Understanding the Elimination Method is crucial for solving problems in engineering, physics, and even AI/ML, where you often deal with multiple unknown values. Engineers use it to design circuits, and data scientists use it in machine learning algorithms to find optimal solutions.
Common Mistakes
MISTAKE: Not multiplying one or both equations by a number to make coefficients opposite or equal before adding/subtracting. | CORRECTION: Always ensure the coefficients of one variable are either the same (for subtraction) or opposites (for addition) before combining equations.
MISTAKE: Making sign errors when subtracting equations or when multiplying an entire equation by a negative number. | CORRECTION: Be very careful with negative signs. When subtracting, remember to change the sign of every term in the second equation.
MISTAKE: Forgetting to substitute the first variable's value back into an original equation to find the second variable. | CORRECTION: After finding one variable, always plug its value back into one of the original equations to solve for 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 = 10 | ANSWER: x = 3, y = 2
QUESTION: Solve using elimination: 2x + 5y = 11 and 3x + 2y = 12 | ANSWER: x = 4, y = 3/5
MCQ
Quick Quiz
Which step is crucial before adding or subtracting equations in the Elimination Method?
Making sure all constants are zero
Ensuring one variable has coefficients that are opposites or identical
Converting equations to fractions
Graphing both lines
The Correct Answer Is:
B
The core idea of elimination is to remove one variable. This is only possible if their coefficients are opposites (to add) or identical (to subtract).
Real World Connection
In the Real World
Imagine a logistics company like Delhivery or Ecom Express managing deliveries. They might use systems of linear equations to figure out the most efficient routes, considering factors like fuel cost per kilometer and average speed on different road types, effectively eliminating variables to find optimal solutions.
Key Vocabulary
Key Terms
LINEAR EQUATION: An equation whose graph is a straight line, involving variables raised to the power of 1 | COEFFICIENT: The numerical factor of a term in an algebraic expression | VARIABLE: A symbol, usually a letter, representing an unknown value | SYSTEM OF EQUATIONS: A set of two or more equations containing common variables | ELIMINATE: To remove or get rid of something
What's Next
What to Learn Next
Now that you've mastered the Elimination Method, explore the Substitution Method and Graphical Method for solving linear equations. Each method offers a different way to tackle similar problems, expanding your problem-solving toolkit!
