Linear Equations in Two Variables

1. Introduction

An equation is a statement of equality involving one or more variables. In earlier classes, you studied linear equations in one variable (e.g., $x + 5 = 10$). Now, we will explore equations involving two variables.

2. Definition

A Linear Equation in Two Variables is an equation that can be written in the form:

$ax + by + c = 0$

Where:

$a, b, c$ are real numbers.
$a \neq 0$ and $b \neq 0$ (Both coefficients cannot be zero).
$x$ and $y$ are variables.

Examples:

$2x + 3y = 5$ (Here $a=2, b=3, c=-5$)
$x - 2y - 3 = 0$ (Here $a=1, b=-2, c=-3$)
$y = 3x$ (Can be written as $3x - y + 0 = 0$)

3. Solution of a Linear Equation

A solution to a linear equation in two variables is a pair of values, one for $x$ and one for $y$, which makes the two sides of the equation equal.

Example: For the equation $2x + y = 7$:

If we put $x = 3$ and $y = 1$: LHS = $2(3) + 1 = 6 + 1 = 7$ = RHS. So, $(3, 1)$ is a solution.
If we put $x = 1$ and $y = 5$: LHS = $2(1) + 5 = 7$ = RHS. So, $(1, 5)$ is also a solution.

Important Property: A linear equation in two variables has infinitely many
                        solutions.
                

Practice Questions - Free Preview

5 Sample Questions

Question 1

The linear equation $2x - 5y = 7$ has:

A. A unique solution

B. Two solutions

C. Infinitely many solutions

D. No solution

Correct Answer: C
A linear equation in two variables represents a line, and a line contains infinitely many points. Hence, it has infinitely many solutions.

Question 2

The equation $x = 7$ can be written in two variables ($x, y$) as:

A. $1 \cdot x + 0 \cdot y - 7 = 0$

B. $1 \cdot x + 1 \cdot y - 7 = 0$

C. $0 \cdot x + 1 \cdot y - 7 = 0$

D. $1 \cdot x + 0 \cdot y + 7 = 0$

Correct Answer: A
Since there is no $y$ term, its coefficient is 0. Moving 7 to LHS gives $x - 7 = 0$, which is $1 \cdot x + 0 \cdot y - 7 = 0$.

Question 3

If $(2, 0)$ is a solution of the linear equation $2x + 3y = k$, then the value of $k$ is:

A. 4

B. 6

C. 5

D. 2

Correct Answer: A
Substitute $x=2, y=0$ in the equation: $2(2) + 3(0) = k \Rightarrow 4 + 0 = k \Rightarrow k = 4$.

Question 4

Any point on the $x$-axis is of the form:

A. $(x, y)$

B. $(0, y)$

C. $(x, 0)$

D. $(x, x)$

Correct Answer: C
On the $x$-axis, the value of the $y$-coordinate (ordinate) is always 0.

Question 5

Which of the following is NOT a linear equation in two variables?

A. $2x + 3y = 7$

B. $x^2 + 2x = 5$

C. $x = 0$

D. $x - 5y = 2$

Correct Answer: B
Equation B has a term $x^2$, making its degree 2. A linear equation must have degree 1.

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Linear Equations in Two Variables - Exam Preparation Strategy

When studying Linear Equations in Two Variables for your final board exams, it is critical to focus on the core concepts and fundamental formulas. Relying strictly on NCERT textbook solutions and practicing previous year questions (PYQs) is the proven methodology for scoring high marks. Avoid rote memorization and instead focus on the logical application of the theories presented in this chapter.

⚠️ Common Mistakes to Avoid

Rushing through the word problems without fully understanding the required parameters.

Failing to revise the critical formulas and definitions on a weekly basis.

Reading the theory without actively practicing Multiple Choice Questions (MCQs) under a time limit.

❓ Frequently Asked Questions

How can I quickly memorize the concepts of Linear Equations in Two Variables?

The most effective way is to create short, handwritten revision notes and continuously test your knowledge using our interactive Mock Tests. Spaced repetition and active recall are much better than passive reading.

What type of questions are most commonly asked from Linear Equations in Two Variables?

Board exams tend to favor conceptual application questions and direct formula-based derivations from the NCERT syllabus. Ensure you have solved every single exercise in the official textbook.

Is reading the NCERT book enough for this chapter?

Yes, the NCERT textbook is the absolute gold standard for board exams. However, to improve your speed and accuracy during the actual exam, you must supplement your reading by solving timed mock tests and objective questions.