Day 5: The Art of Elimination | Secondary Stage (Grades 9 & 10) | Apex Institute of Maths and Sciences

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Day 5: The Art of Elimination | Secondary Stage (Grades 9 & 10) | Apex Institute of Maths and Sciences

Day 5: The Art of Elimination 💥

Secondary Stage (Grades 9 & 10) | Apex Institute of Maths and Sciences

🎯 1. Concept: Destroying Variables

Welcome to Day 5! Yesterday, we solved systems of equations using Substitution. That method is great, but sometimes it takes too long. Today, we learn the hacker’s shortcut: The Elimination Method.

The goal of Elimination is to stack two equations on top of each other and literally ADD or SUBTRACT them together so that one of the variables (+y and -y) completely destroys itself and turns into zero!

💡 2. The Elimination Magic

Look at this system:

Eq 1: 2x + y = 10
Eq 2: 3x – y = 5

  • Step 1 (Stack & Add): Notice we have a ‘+y’ in the top and a ‘-y’ in the bottom. If we add the two equations straight down, the y’s will cancel out (+y – y = 0).
  • Addition: (2x + 3x) = 5x. The y’s are gone. (10 + 5) = 15.
    New Equation: 5x = 15
  • Step 2 (Solve): Divide by 5. x = 3.
  • Step 3 (Plug it back in): Put x=3 into the first equation: 2(3) + y = 10. So, 6 + y = 10. Therefore, y = 4.

What if they don’t match? If you have +2y and +y, you can multiply the entire bottom equation by -2 first, THEN add them to eliminate!

🌍 3. Math in Our Daily Life

Elimination is used to quickly find hidden costs or balance chemical reactions!

Scenario 1 (The Bakery): You buy 2 coffees and 1 donut for ₹120 (2c + d = 120). The next day, you buy 2 coffees and 3 donuts for ₹200 (2c + 3d = 200). If you just SUBTRACT yesterday’s receipt from today’s, the cost of the coffees cancels out! You immediately see that the 2 extra donuts cost ₹80. One donut is ₹40.

Scenario 2 (Chemistry): When balancing chemical equations, scientists use elimination to ensure the exact same number of atoms (like Oxygen and Hydrogen) are on both sides of the reaction.

📝 4. Analytical Tasks

Open your math journal and complete these algebraic challenges:

  • Task A: The Setup: Stack these equations: (4x + 2y = 20) and (x – 2y = 5). Add them together. What is your new equation? Solve for x.
  • Task B: The Multiplier: You have (x + y = 10) and (2x + 3y = 26). You want to eliminate the ‘x’. What number must you multiply the ENTIRE top equation by before adding them?

✅ 5. Day 5 Application Test

Can you eliminate the unknowns? This quiz gets progressively harder. Select your answers below and click submit.

Easy
1. What is the main goal of the Elimination Method?
Solution: The goal is to “eliminate” a variable so you are left with a simple equation that has only one letter to solve.
Easy
2. If you add (+y) and (-y) together, what is the result?
Solution: A positive y and a negative y are opposites. They cancel each other out to equal exactly 0!
Easy
3. To eliminate a variable by adding two equations, the numbers attached to the variable (coefficients) must be…
Solution: To cancel out to zero when adding, the numbers must be opposites (e.g., +3 and -3).
Easy
4. Add these two expressions together: 5x + (-5x) = ?
Solution: 5 minus 5 is zero. The x is completely eliminated!
Medium
5. Look at this system: Eq 1 is (2x + y = 12). Eq 2 is (3x – y = 8). If you add the equations together, what is the value of x?
Solution: The +y and -y cancel out. Add the rest: 2x + 3x = 5x. 12 + 8 = 20. New equation: 5x = 20. Divide by 5 to get x = 4.
Medium
6. If you found out that x = 4 in the equation (x + y = 10), what is the value of y?
Solution: Plug the 4 back in: 4 + y = 10. Subtract 4 from both sides. y = 6.
Medium
7. Eq 1 is (2x + y = 5). Eq 2 is (4x + y = 9). If we just ADD them together, we get (6x + 2y = 14). Did we successfully eliminate a variable?
Solution: No! We still have an ‘x’ and a ‘y’. To eliminate the y, we should have multiplied one of the equations by -1 before adding, or simply subtracted the equations.
Medium
8. You want to eliminate the ‘y’ variable. Eq 1 has (+y). Eq 2 has (+2y). What should you multiply Eq 1 by to make the y’s exact opposites?
Solution: You want the top ‘y’ to become ‘-2y’ so it cancels out with the ‘+2y’. Therefore, multiply the entire top equation by -2.
Hard
9. 🧠 Puzzle: A test has 20 questions total. Correct answers (x) give you points, wrong answers (y) take away points. Total questions: (x + y = 20). Your score equation: (3x – y = 32). How many questions did you get CORRECT (x)?
Solution: Add the two equations! The +y and -y cancel. (x + 3x) = 4x. (20 + 32) = 52. So, 4x = 52. Divide by 4 to get x = 13 correct answers.
Hard
10. 🧠 Puzzle: You buy 2 apples and 3 bananas for ₹35 (2a + 3b = 35). Later, you buy 2 apples and 5 bananas for ₹45 (2a + 5b = 45). If you subtract the first equation from the second to eliminate the apples, what is the cost of ONE banana (b)?
Solution: Subtract Eq 1 from Eq 2: (2a – 2a = 0). (5b – 3b = 2b). (45 – 35 = 10). New equation: 2b = 10. Divide by 2: b = 5. One banana costs ₹5.
⚠️ Please answer all 10 questions before submitting!
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