Linear Equations in Two Variables – Easy Guide

Linear Equations in Two Variables – Easy Guide

Linear Equations in Two Variables – Full Guide for Beginners to Master Concepts Fast!

Have you ever tried to figure out how two unknown values are connected? Imagine you’re buying pens and notebooks, and you only know the total cost—but not the individual prices. Sounds tricky, right?

This is exactly where Linear Equations in Two Variables come to the rescue.

These equations are not just a chapter in your syllabus—they are powerful tools used in real life, from budgeting and business calculations to engineering and data science. Once you understand them, solving problems becomes faster, easier, and even fun!

Let’s break it down step-by-step in the simplest way possible.

Sets Explained for POLYCET 2026 Master Concepts Fast & Score High

What is a Linear Equation in Two Variables?

  An equation of the form ax + by + c = o where a, b, c are real numbers and (a² + b² ≠ 0) is called a linear equation in two variables.

Pair of Linear equations in two Variables:

Two linear equations in two variables of the same type are called a pair of linear equations in two variables.

a₁x + b₁y + c₁ = 0 (a₁² + b₁² ≠ 0), a₂x + b₂ y + c₂ = 0 (a₂² + b₂²≠0); a₁, a₂, b₁, b₂, c₁, c₂ are real numbers

Graphical Representation

When two lines are drawn in the same plane, only one of the following three situations is possible:

1. The two lines may intersect at one point  .
2. The two lines may not intersect, i.e., they are parallel  .
3. The two lines may be coincident. (actually, both are the same) 

Types of Solutions

Let a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 form a pair of linear equations in two variables, then the following situations can arise:

Three methods can solve a pair of linear equations

1. Graphical method
2. Substitution method
3. Elimination method
4. Cross multiplication method

Solved Questions with Answers

1. If the system of equations 2x + 3y = 7, 2ax + (a + b)y = 28 has infinitely many solutions, then1) a = 2b     (2) b = 2a      (3) a + 2b = 0         (4) 2a + b = 0

Answer:   (2)

Solution: 

Given Equations are 2x + 3y = 7, 2ax + (a + b)y = 28 has infinitely many solutions

⇒ a₁/a₂  = b₁/b₂ = c₁/c₂

⇒ a/2a  = 3/a + b = – 7/– 28 = 1/4

⇒ 1/a = 1/4 ⇒ a = 4

⇒   3/a + b = 1/4

a + b = 12

4 + b = 12 ⇒ b = 8

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