Linear Equations in Two Variables — Lesson
1) Hook — A Fun Real-Life Story
Imagine you are at an Indian sweet shop in Chandni Chowk, Delhi. You want to buy laddoos and barfis. Each laddoo costs ₹10 and each barfi costs ₹15. You have ₹150 with you. How many laddoos and barfis can you buy so that you spend exactly ₹150? This situation can be represented by a linear equation in two variables — the number of laddoos and barfis.
2) Core Concepts — Understanding Linear Equations in Two Variables
A linear equation in two variables is an equation that can be written in the form:
where a, b, and c are real numbers and a and b are not both zero. Here, x and y are variables representing unknown values.
Example: 3x + 4y = 12
What does this mean? For every pair of values (x, y) that satisfy this equation, the equation holds true.
Finding Solutions
To find solutions, choose a value for one variable and solve for the other.
| x | y (from 3x + 4y = 12) |
|---|---|
| 0 | 3 (since 3*0 + 4y = 12 → y = 3) |
| 4 | 0 (3*4 + 4y = 12 → y = 0) |
| 2 | 1.5 (3*2 + 4y = 12 → 6 + 4y = 12 → y = 1.5) |
Each pair (x, y) is called a solution of the equation.
Graphical Representation
All solutions of a linear equation in two variables lie on a straight line when plotted on the Cartesian plane.
Example: For 3x + 4y = 12, points (0,3), (4,0), and (2,1.5) lie on the same straight line.
3) Key Formulas / Rules
General form of linear equation in two variables:
ax + by = c
where a ≠ 0 or b ≠ 0
To find y in terms of x:
y = (c - ax) / b
(provided b ≠ 0)
Number of solutions:
- Infinite solutions (all points on the line)
- No solution if lines are parallel (in system of equations)
- Exactly one solution if lines intersect at one point (in system of equations)
4) Did You Know?
Linear equations in two variables are used in Indian agriculture to calculate the right combination of fertilizers. For example, if a farmer wants to mix two types of fertilizers costing ₹x and ₹y per kg to get a mixture costing ₹z per kg, the problem can be modelled as a linear equation!
5) Exam Tips — Avoid These Common Mistakes!
- Don’t forget: Both variables must be present for it to be a linear equation in two variables.
- Check coefficients: If a or b is zero, it reduces to a linear equation in one variable.
- Substitution carefully: When finding solutions, substitute values carefully to avoid arithmetic errors.
- Graphing: Plot at least two points correctly to draw the line accurately.
- Exam Pattern: Questions may ask to find solutions, write equations from word problems, or plot graphs.
- Mnemonic to remember general form: "All Boys Can" → a x + b y = c
Linear Equations in Two Variables — Mcq
Linear Equations in Two Variables — Mnemonic
Mnemonic 1: "LINEAR" for Remembering the Form of Linear Equations 📐
- L = Linear (straight line)
- I = In (in two variables)
- N = Numbers (coefficients like a, b, c)
- E = Equation (ax + by = c)
- A = Always degree 1
- R = Represent straight lines on graph
Remember: "L-I-N-E-A-R = Straight Line Equation!" 😎
Mnemonic 2: Hindi Phrase to Recall Variables and Coefficients 🇮🇳📝
"अच्छे बच्चे पढ़ते हैं, चलो जोड़ते हैं!"
- अच्छे (a) = Coefficient of x
- बच्चे (b) = Coefficient of y
- पढ़ते (p) = Plus sign (+)
- हैं (h) = Equals (=)
- चलो (c) = Constant term
- जोड़ते (j) = Join variables and constants properly
So, "a x + b y = c" is easy to remember with this fun line! 😄
Mnemonic 3: Rhyming Trick for Solutions of Linear Equations 🎶
"Two variables, one line, solve for x and y just fine!"
- Remember: ax + by = c represents a straight line.
- Every solution is a point (x, y) on that line.
- Pick any value for x or y, solve for the other — easy as pie! 🥧
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