Day 17: Linear Equations in Two Variables | Secondary Stage Mathematics | Apex Institute of Maths and Sciences

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Day 17: Linear Equations in Two Variables | Secondary Stage Mathematics | Apex Institute of Maths and Sciences

Day 17: Linear Equations in Two Variables | Secondary Stage Mathematics | Apex Institute of Maths and Sciences

Level 1: The Quest

The Mystery of the Variables

Imagine you are a detective. You have two clues (equations) and two suspects ($x$ and $y$). A Linear Equation in Two Variables is an equation that can be written in the form $ax + by + c = 0$. Our quest today is to find the exact values of $x$ and $y$ that make both equations true at the same time!

Example System:
1) $x + y = 10$
2) $x – y = 2$
The solution is $(6, 4)$ because $6+4=10$ and $6-4=2$.
Level 2: Power-Ups

Battle Tactics: 3 Ways to Solve

1. Substitution: Express one variable in terms of the other from one equation and “plug” it into the second.
2. Elimination: Multiply equations to make the coefficients of one variable equal, then add or subtract to “kill” that variable.
3. Cross-Multiplication: Use the “Magic Formula”: $$\frac{x}{b_1c_2 – b_2c_1} = \frac{y}{c_1a_2 – c_2a_1} = \frac{1}{a_1b_2 – a_2b_1}$$
Pro-Tip: Always check the ratio $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ first to ensure a Unique Solution exists!
Level 3: Mini-Boss Battles

Real-World Combat

🚀 Scenario 1: The Ticket Counter
A movie theater sells adult tickets ($x$) for ₹200 and child tickets ($y$) for ₹100. If 50 people went and total collection was ₹8000, we solve:
$x + y = 50$ and $200x + 100y = 8000$.
⚡ Scenario 2: Speed vs Stream
A boat travels upstream and downstream. Let boat speed be $x$ and stream be $y$.
Downstream speed $= x+y$; Upstream speed $= x-y$.
Level 4: Home Quests

XP Missions

Task 1: The Kitchen Inventory
Find two different types of items (e.g., Spoons and Forks). Count total items. Create two equations based on their weights or count ratios with a parent’s help.
Task 2: The Age Riddle
Ask your father his age. Create a system of equations: “Today, the sum of our ages is S. In 5 years, Dad will be 3 times as old as me.” Solve for $x$ and $y$.
Final Boss: Practice Test

1. Which of the following is a linear equation in two variables? EASY

Magic Solution: A is correct because the power of variables $x$ and $y$ is 1, and it contains exactly two variables.

2. If $x=2, y=3$ is a solution of $2x + ky = 13$, find $k$. EASY

Magic Solution: Substitute $x=2, y=3$: $2(2) + k(3) = 13 \Rightarrow 4 + 3k = 13 \Rightarrow 3k = 9 \Rightarrow k=3$.

3. The graph of $x = 5$ is a line: EASY

Magic Solution: $x = a$ is always a vertical line parallel to the y-axis.

4. For what value of $k$ do the equations $3x-y+8=0$ and $6x-ky+16=0$ represent coincident lines? EASY

Magic Solution: For coincidence, $\frac{3}{6} = \frac{-1}{-k} \Rightarrow \frac{1}{2} = \frac{1}{k} \Rightarrow k=2$.

5. Solve by elimination: $x+y=5$ and $x-y=1$. MODERATE

Magic Solution: Adding both: $2x = 6 \Rightarrow x=3$. Putting $x=3$ in first: $3+y=5 \Rightarrow y=2$.

6. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the system has: MODERATE

Magic Solution: This condition represents parallel lines which never intersect.

7. The sum of digits of a two-digit number is 9. If 27 is added to it, digits reverse. The number is: MODERATE

Magic Solution: $3+6=9$ and $36+27=63$ (reversed). Equation form: $(10x+y)+27 = 10y+x$.

8. In $ax+by+c=0$, if $c=0$, the line always passes through: MODERATE

Magic Solution: Substituting $(0,0) \Rightarrow a(0)+b(0)+0=0$. It satisfies the equation.

9. Solve for $x$ and $y$: $\frac{2}{x} + \frac{3}{y} = 13$ and $\frac{5}{x} – \frac{4}{y} = -2$. COMPLEX

Magic Solution: Let $1/x=u, 1/y=v$. Equations: $2u+3v=13, 5u-4v=-2$. Solving gives $u=2, v=3 \Rightarrow x=1/2, y=1/3$.

10. Aruna has only ₹1 and ₹2 coins. Total coins = 50, Total money = ₹75. Number of ₹1 and ₹2 coins are: COMPLEX

Magic Solution: $x+y=50$ and $1x+2y=75$. Subtracting gives $y=25$, then $x=25$.
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