Day 2: Crack the Code (Linear Equations) 🕵️‍♂️| Secondary Stage | Apex Institute of Maths and Sciences

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Day 2: Crack the Code (Linear Equations) 🕵️‍♂️

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

🎯 1. Concept: The Balancing Act

Welcome back! Yesterday we learned about variables (like x). Today, we are going to become mathematical detectives and find out exactly what number x is hiding.

An Equation is like a weighing scale. The equal sign (=) is the center. Whatever you do to one side of the equation, you MUST do to the other side to keep it balanced.

Your main mission is always the same: Isolate the variable. You want to get ‘x’ all by itself on one side of the equal sign.

💡 2. Inverse Operations

To move a number away from ‘x’, use its opposite (inverse) operation:

  • Opposite of Addition is Subtraction:
    If x + 5 = 12, subtract 5 from both sides. (x = 7)
  • Opposite of Subtraction is Addition:
    If y – 4 = 10, add 4 to both sides. (y = 14)
  • Opposite of Multiplication is Division:
    If 3m = 15 (which means 3 times m), divide both sides by 3. (m = 5)

Two-Step Example: 2x + 1 = 9. First, subtract 1 (2x = 8). Then, divide by 2 (x = 4).

🌍 3. Math in Our Daily Life

Equations help us solve real-world mysteries!

Scenario 1 (Shopping): You have exactly ₹500. You buy a shirt for ₹300 and 2 identical pairs of socks. How much did one pair of socks cost?
Equation: 2x + 300 = 500. Subtract 300 (2x = 200). Divide by 2 (x = 100). The socks cost ₹100 each!

Scenario 2 (Travel): A taxi charges a ₹50 booking fee plus ₹10 for every kilometer. If your total bill is ₹150, how far did you travel?
Equation: 10x + 50 = 150. Subtract 50 (10x = 100). Divide by 10 (x = 10). You traveled 10 km!

📝 4. Analytical Tasks

Open your math journal and complete these algebraic challenges:

  • Task A: Build an Equation: Write an equation for this sentence: “Three times a number, minus four, equals fourteen.” Then solve for the number.
  • Task B: Solve it: Solve the equation: 5p – 7 = 18. Show your steps (what did you add first? What did you divide by?).
  • Task C: Real World: Create your own real-life word problem that matches the equation 4x + 20 = 100.

✅ 5. Day 2 Practice Test

Can you crack the code? Select the correct answers below and click submit to check your score and view the step-by-step solutions.

1. What is the ultimate goal when solving a linear equation?
Solution: The goal is always to get the variable isolated on one side of the equal sign (e.g., x = 5).
2. Solve for x: x + 7 = 15
Solution: Subtract 7 from both sides. x = 15 – 7. Therefore, x = 8.
3. Solve for y: y – 4 = 10
Solution: Use the inverse of subtraction. Add 4 to both sides. y = 10 + 4. Therefore, y = 14.
4. What is the inverse (opposite) operation of multiplication?
Solution: Division is the opposite of multiplication. They cancel each other out!
5. Solve for m: 3m = 21
Solution: 3m means 3 multiplied by m. To isolate m, divide both sides by 3. m = 21 / 3. So, m = 7.
6. Solve for a: a / 2 = 6
Solution: The inverse of division is multiplication. Multiply both sides by 2. a = 6 × 2. Therefore, a = 12.
7. Two-step equation! Solve: 2x + 1 = 9
Solution: Step 1: Subtract 1 from both sides (2x = 8). Step 2: Divide both sides by 2 (x = 4).
8. Solve for p: 5p – 2 = 18
Solution: Step 1: Add 2 to both sides (5p = 20). Step 2: Divide both sides by 5 (p = 4).
9. You buy a book for ₹20 and some pens for ₹5 each. You spend ₹45 in total. Which equation represents this if ‘x’ is the number of pens?
Solution: The ₹20 is a one-time constant. The ₹5 multiplies by the number of pens (x). So, 5x + 20 = 45.
10. Complete the Golden Rule of Equations: “Whatever you do to the left side of the equation…”
Solution: You must do the exact same thing to the right side to keep the “scale” perfectly balanced!
⚠️ Please answer all 10 questions before submitting!

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