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I. Chapter Summary

This chapter introduces students to linear equations involving two variables, which are algebraic expressions representing straight lines on a coordinate plane. The general form of a linear equation is $ax + by + c = 0$, where a, b, and c are real numbers and a and b are not both zero. Students learn how to plot these equations on a graph, interpret solutions geometrically, and understand that each linear equation in two variables has infinitely many solutions. This foundational chapter is crucial for algebraic reasoning and analytical geometry in higher classes.

II. Key Concepts Covered

Concept	Explanation
Linear Equation in Two Variables	An equation of the form $ax + by + c = 0$, where $a, b, c in mathbb{R} text{ and } a, b neq 0$ simultaneously.
Solution of a Linear Equation	A pair of values (x, y) that satisfies the equation. There are infinitely many solutions.
Graph of a Linear Equation	A straight line formed by plotting at least two or more solution points of the equation.
Standard Form	The expression of the equation as $ax + by + c = 0$.
Geometrical Representation	Each solution corresponds to a point on the line; the line represents all solutions.
Table of Values	Used to calculate and organize pairs of x and y that satisfy the equation for plotting.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. Which of the following is a linear equation in two variables?
   a) $x^2 + y = 2$
   b) $x + y = 7$ ✔️
   c) $x^2 – y^2 = 5$
   d) $xy = 6$
2. The graph of a linear equation in two variables is always a:
   a) curve
   b) circle
   c) straight line ✔️
   d) parabola
3. Which of the following is not a solution of the equation x + 2y = 6?
   a) (2, 2)
   b) (4, 1)
   c) (0, 3) ✔️
   d) (6, 0)
4. The number of solutions of a linear equation in two variables is:
   a) 1
   b) 2
   c) 3
   d) infinitely many ✔️
   (PYQ 2019)

(B) Short Answer Questions (2/3 Marks)

1. Find four solutions of the equation $2x + y = 7$.
2. Check whether the point (1, 2) lies on the line represented by $3x + y = 5$.
3. Write the linear equation representing the statement: “The sum of a number x and twice y is 10.”
4. Draw the graph of the equation $x + y = 4$ using any two points. (PYQ 2020)

(C) Long Answer Questions (5 Marks)

1. Draw the graph of $2x + y = 6$. Find the value of y when $x = 0, 1$, and 2.
2. A number is 5 more than three times another number. Represent this using a linear equation and draw its graph.
3. For the equation $x – 2y = 4$, prepare a table of three solutions and draw its graph. Also identify the quadrant in which each point lies.
4. Write a real-life situation which can be represented by a linear equation in two variables and solve it graphically.

(D) HOTS (Higher Order Thinking Skills)

1. Can two different linear equations have exactly one common solution? If yes, illustrate with an example and explain.
2. A line passes through points (–1, 2) and (3, 4). Find its equation and justify why it’s linear.

IV. Key Formulas/Concepts

Concept	Key Point / Formula
General Form	$ax + by + c = 0$
Solution of Linear Equation	Any pair (x, y) that satisfies the equation
Graph of Linear Equation	A straight line representing all solutions
Table of Values	Used to plot points on graph paper
Infinitely Many Solutions	Linear equation in two variables has infinite solutions
To Plot	Choose values of x (or y), compute y (or x), and plot those points

V. Deleted Portions (CBSE 2025–2026)

 No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Chapter	Estimated Marks	Types of Questions Typically Asked
Linear Equations in Two Variables	6–8 Marks	1 MCQ, 1 Short Answer, 1 Graph-Based Long Answer, 1 Conceptual/HOTS

VII. Previous Year Questions (PYQs)

Marks	Question	Year
1 mark	The number of solutions of a linear equation in two variables is?	PYQ 2019
2 marks	Plot graph of $x + y = 4$ using any two points.	PYQ 2020
3 marks	Write four solutions for the equation $2x + y = 6$.	PYQ 2018
5 marks	Draw the graph of $y = 3x – 1$. Find coordinates of any point lying on it.	PYQ 2020

VIII. Real-World Application Examples

• Budget Planning: “Sum of price of x kg apples and y kg mangoes = ₹500” → x + y = constant.
• Transportation Cost: Distance vs fare rates for different vehicles represented through linear equations.
• Business & Finance: Representing profit = revenue – expenses using linear equations.
• Mixture Problems: Combining different concentrations or quantities in chemistry, economics.

IX. Student Tips & Strategies for Success

 Time Management

• 3-day plan:
  Day 1: Concept + Examples
  Day 2: Graph plotting + Tables
  Day 3: Word problems + PYQs

 Exam Preparation

• Always label axes, points, and lines in graph questions.
• Use at least 3 points for accurate graphing.

 Stress Management

• Practice in a relaxed environment with graph paper.
• Create your own situations and try converting them into equations.

X. Career Guidance & Exploration

• For Classes 9–10:
  ➤ Strong algebraic base is needed for higher classes
  ➤ Recommended preparation for NTSE, SOF Olympiads, RMO
• For Classes 11–12 & Beyond:
  ➤ Careers using linear algebra:
  • Data Science
  • Economics
  • Engineering (especially Electrical & Computer)
  • Logistics & Operations Research
  ➤ Important for JEE, CUET, NDA, and competitive exams

XI. Important Notes

 Linear equations represent real-world relationships — practice formulating them from verbal problems.
 Graph accuracy and labeling are critical in board exams.
 Understand that every (x, y) solution lies on the line and every point on the line is a solution.
 Revise standard forms and ensure clarity on how to create tables for graphing.