Day 14: Linear Equations | Middle Stage Mathematics | Apex Institute of Maths and Sciences

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Day 14: Linear Equations | Middle Stage Mathematics

Apex Institute of Maths and Sciences

🚀 Level 1: The Quest (Concept)

Imagine a weighing scale. For the scale to stay balanced, both sides must have the exact same weight. A Linear Equation with variables on both sides is just like that! Our mission is to move all the “mystery boxes” (variables) to one side and all the “weights” (numbers) to the other side to find the secret value.

Balancing Act:
\( 2x + 5 = x + 10 \)
To solve, we subtract \(x\) from both sides:
\( x + 5 = 10 \)
Then subtract 5:
\( x = 5 \)

⚡ Level 2: Power-Ups (Tools & Methods)

The Golden Rule: Whatever you do to the left side, you MUST do to the right side!

Use the “Variables to the Left, Numbers to the Right” strategy:

Step Action
1. Simplify Clear brackets using distribution.
2. Move Variable Add/Subtract variable terms to collect them on one side.
3. Isolate Add/Subtract constants to move them to the other side.

⚔️ Level 3: Mini-Boss Battles (Applications)

Scenario 1: The Cab Challenge
Cab A charges ₹50 plus ₹10 per km. Cab B charges ₹20 plus ₹15 per km. When is the cost the same?
Equation: \( 10x + 50 = 15x + 20 \)
Scenario 2: Gym Membership
Gym X has a ₹500 joining fee and ₹100 monthly. Gym Y has no fee but ₹200 monthly. After how many months are they equal?
Equation: \( 100x + 500 = 200x \)

🏠 Level 4: Home Quests (Activities)

Quest 1: The Shopping Detective
Go to the kitchen with a parent. Find two different sized packets of the same item (e.g., Biscuits). Calculate the price per gram and write an equation to see at what weight both would cost the same.
Quest 2: Age Riddle Maker
Create an equation based on your age and your parent’s age. Example: “In how many years will my dad’s age be double mine?” Solve it together!

👹 Final Boss: Practice Test

1. Solve for \(x\): \( 3x = x + 10 \) EASY

Magic Solution: Subtract \(x\) from both sides: \( 2x = 10 \). Divide by 2: \( x = 5 \). ✔

2. Solve: \( 5y – 4 = 2y + 5 \) EASY

Magic Solution: \( 5y – 2y = 5 + 4 \Rightarrow 3y = 9 \Rightarrow y = 3 \). ✔

3. What is the first step to solve \( 4x + 2 = 6x – 8 \)? EASY

Magic Solution: To isolate \(x\), you should gather variable terms on one side and constants on the other. ✔

4. Solve: \( 10 – 2x = x – 5 \) EASY

Magic Solution: \( 10 + 5 = x + 2x \Rightarrow 15 = 3x \Rightarrow x = 5 \). ✔

5. Solve: \( 2(x + 3) = x + 10 \) MODERATE

Magic Solution: Expand: \( 2x + 6 = x + 10 \). Subtract \(x\): \( x + 6 = 10 \). Result: \( x = 4 \). ✔

6. A number doubled plus 5 is the same as the number plus 12. Find the number. MODERATE

Magic Solution: Equation: \( 2x + 5 = x + 12 \). Subtract \(x\): \( x + 5 = 12 \). Result: \( x = 7 \). ✔

7. Solve: \( \frac{x}{2} + 4 = x – 2 \) MODERATE

Magic Solution: Multiply all by 2: \( x + 8 = 2x – 4 \). Rearrange: \( 8 + 4 = 2x – x \). \( x = 12 \). ✔

8. If \( 3(2x – 1) = 4x + 7 \), find \(x\). MODERATE

Magic Solution: \( 6x – 3 = 4x + 7 \Rightarrow 2x = 10 \Rightarrow x = 5 \). ✔

9. Solve: \( 0.5x + 3 = 0.2x + 6 \) COMPLEX

Magic Solution: \( 0.5x – 0.2x = 6 – 3 \Rightarrow 0.3x = 3 \Rightarrow x = 10 \). ✔

10. The perimeter of a rectangle is 40. Length is \( 2x+2 \) and width is \( x+3 \). Find \(x\). COMPLEX

Magic Solution: \( 2(L+W) = 40 \Rightarrow (2x+2) + (x+3) = 20 \Rightarrow 3x + 5 = 20 \Rightarrow 3x = 15 \Rightarrow x = 5 \). ✔

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