Day 4: The Art of Substitution | Secondary Stage | Apex Institute of Maths and Sciences

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Day 4: The Art of Substitution | Secondary Stage | Apex Institute of Maths and Sciences

Day 4: The Art of Substitution 🕵️‍♀️🧩

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

🎯 1. Concept: Systems of Equations

Welcome to Day 4! Until now, we have solved for just one unknown variable (like finding ‘x’). But what if a mystery has TWO unknowns—like trying to find the price of an apple (a) AND a banana (b) at the same time?

If you have two unknowns, you MUST have two equations to solve them. When we group two or more equations together, it is called a System of Equations.

Yesterday, we graphed lines. The solution to a system is simply the exact coordinate point where the two lines intersect on a graph! Today, we will find that exact point without drawing a thing.

💡 2. The Substitution Method

Think of substitution like a sports game. If a player gets injured, you swap them out for someone of the exact same value. Let’s solve this system:

Equation 1: y = 2x
Equation 2: x + y = 12

  • Step 1: Equation 1 tells us that ‘y’ is the exact same thing as ‘2x’. They are equal players.
  • Step 2: Go to Equation 2, take out the ‘y’, and substitute ‘2x’ in its place.
    New Equation: x + (2x) = 12
  • Step 3: Solve it! 3x = 12, so x = 4.
  • Step 4: Now that we know x is 4, put it back into the first equation to find y: y = 2(4), so y = 8.

The solution is the coordinate point (4, 8).

🌍 3. Math in Our Daily Life

Systems of equations are how businesses make decisions and solve complex puzzles!

Scenario 1 (The Farm Puzzle): A farmer sees chickens (2 legs) and cows (4 legs). He counts exactly 10 heads in total, and 26 legs in total. How many of each animal are there? By creating a system (c + k = 10, and 4c + 2k = 26), he can use substitution to find out without counting them individually!

Scenario 2 (Business Break-Even): A company spends ₹500 to make a product, and sells it for ₹1000. They use a system of equations to find the “Break-Even Point”—the exact moment where their Cost line crosses their Profit line on a graph.

📝 4. Analytical Tasks

Open your math journal and complete these algebraic challenges:

  • Task A: The Swap: You are given: y = x + 3, and 2x + y = 18. Rewrite the second equation by substituting the first one into it. (Don’t solve it, just write the new equation).
  • Task B: Solve the Puzzle: The sum of two secret numbers is 20. The difference between them is 4. What are the two numbers? (Hint: Set up x+y=20 and x-y=4).
  • Task C: Real World: Create your own word problem about buying two different items at a store that would require a system of equations to solve.

✅ 5. Day 4 Advanced Application Test

This is a true test of logical reasoning. Grab a pen and paper—you will need to translate these real-world scenarios into math before you can answer them! Select the correct answers below and click submit.

1. If you graph a System of Linear Equations, what does the “solution” to the problem look like visually?
Solution: The solution to a system is the exact intersection point of the lines. It is the only (x,y) coordinate that works for BOTH equations!
2. You know that y = x + 5. The second equation is 2x + y = 20. If you use the substitution method, what will your new, combined equation look like?
Solution: You take the “y” out of the second equation, and drop “(x + 5)” right into its place.
3. 🧠 Puzzle: The sum of two numbers is 30. Their difference is 10. What is the LARGER number?
Solution: Equations: x + y = 30, and x – y = 10. From eq 2: x = y + 10. Substitute: (y + 10) + y = 30 -> 2y = 20 -> y = 10. If y is 10, x must be 20!
4. 🧠 Puzzle: A notebook costs ₹20 more than a pen. You buy 2 pens and 1 notebook for exactly ₹50. What is the cost of ONE pen?
Solution: Notebook (n) = Pen (p) + 20. Equation: 2p + n = 50. Substitute: 2p + (p + 20) = 50 -> 3p = 30. One pen is ₹10!
5. Why do we even use the substitution method? What is the main mathematical goal?
Solution: You can’t solve an equation with both an ‘x’ and a ‘y’ in it. Substitution turns it into an equation with ONLY ‘x’s or ONLY ‘y’s.
6. 🧠 Puzzle: You have a pocket full of 10-rupee coins and 5-rupee coins. You have 8 coins in total, and their total value is ₹65. How many 10-rupee coins do you have?
Solution: Let t=tens, f=fives. (t + f = 8), so f = 8 – t. Value eq: 10t + 5f = 65. Sub: 10t + 5(8 – t) = 65 -> 10t + 40 – 5t = 65 -> 5t = 25. You have 5 ten-rupee coins!
7. 🧠 Puzzle: A rectangle’s length is twice its width (L = 2W). The perimeter of the rectangle is 30 cm. What is the width?
Solution: Perimeter = 2L + 2W. So, 2L + 2W = 30. Substitute L: 2(2W) + 2W = 30 -> 4W + 2W = 30 -> 6W = 30 -> W = 5 cm.
8. 🧠 Puzzle: Rohan is 5 years older than his sister. The sum of their ages is 25. How old is Rohan?
Solution: Rohan (R) = Sister (S) + 5. Equation: R + S = 25. Sub: (S + 5) + S = 25 -> 2S = 20 -> Sister is 10. That means Rohan is 15!
9. If you are solving a system and you get a weird, impossible result like “0 = 5” where all the variables disappear, what does that mean?
Solution: An impossible statement like 0 = 5 means the system has No Solution. Visually, the lines are parallel and will never intersect!
10. In the system: x = 4y, and x + y = 10… what is the value of y?
Solution: Substitute 4y in place of x. (4y) + y = 10 -> 5y = 10 -> y = 2.
⚠️ Please answer all 10 questions before submitting!
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