Linear Equations — MCQ Practice
Pair of Linear Equations in Two Variables — Lesson
1) Hook — A Fun Real-Life Story
Imagine you and your friend go to a sweet shop in Delhi. You buy 2 laddoos and 3 barfis for ₹70. Your friend buys 3 laddoos and 2 barfis for ₹65. How can you find the price of one laddoo and one barfi? This is where Pair of Linear Equations in Two Variables helps us solve such problems easily!
2) Core Concepts
Definition: A pair of linear equations in two variables is a set of two linear equations, each in the form ax + by = c, where x and y are variables and a, b, c are constants.
General form:
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
Example: Using the sweet shop problem, let x = price of one laddoo, y = price of one barfi.
| Items | Equation 1 (₹70) | Equation 2 (₹65) |
|---|---|---|
| Laddoos | 2x | 3x |
| Barfis | 3y | 2y |
| Total Cost | 70 | 65 |
So, the pair of linear equations is:
2x + 3y = 70
3x + 2y = 65
Solving these equations (by substitution, elimination, or cross-multiplication) will give the price of laddoo and barfi.
3) Key Formulas / Rules
General form of pair of linear equations:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Condition for the nature of solutions:
| Condition | Interpretation |
|---|---|
| (a₁/b₁) ≠ (a₂/b₂) | Unique solution (intersecting lines) |
| (a₁/b₁) = (a₂/b₂) ≠ (c₁/c₂) | No solution (parallel lines) |
| (a₁/b₁) = (a₂/b₂) = (c₁/c₂) | Infinite solutions (coincident lines) |
Cross-Multiplication Method: To solve
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Use formulas:
x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁)
y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁)
4) Did You Know?
Linear equations have been used since ancient times! The famous Indian mathematician Brahmagupta (7th century) worked on solving linear and quadratic equations. The word "algebra" itself comes from the Arabic word "al-jabr", introduced by Persian mathematician Al-Khwarizmi, who was influenced by Indian mathematics.
5) Exam Tips
- Always write equations clearly: Identify variables and constants carefully from word problems.
- Check coefficients: Avoid sign mistakes (+/-) while forming equations.
- Use elimination or cross-multiplication: These methods are faster and less error-prone for board exams.
- Remember the conditions for solutions: Write the nature of solution (unique, no solution, infinite) based on coefficient ratios.
- Practice word problems from NCERT and previous years: Most questions follow similar patterns.
- Mnemonic to remember ratios condition: "If Ratios Equal, Lines Parallel or Coincident; Else, Intersect."
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