CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables Notes

Pair of Linear Equations in Two Variables is Chapter 3 of Class 10 Mathematics. It extends your understanding of linear equations to a system of two equations in two unknowns. You will learn to solve them using graphical, substitution, elimination, and cross-multiplication methods, and interpret their solutions geometrically.

Key Topics Covered

•         Linear Equations in Two Variables — Basics

•         Graphical Method of Solution

•         Algebraic Methods: Substitution, Elimination, Cross-Multiplication

•         Nature of Solutions — Consistency Conditions

•         Word Problems on Linear Equations

•         Quick Formula Summary & Board Exam Practice

1.  Linear Equations in Two Variables

A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, c are real numbers and a ≠ 0, b ≠ 0. A pair of such equations forms a system of linear equations.

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Nature of Solutions — Consistency Conditions

The nature of the solution depends on the ratio of the coefficients. Three cases are possible:

Key: Compare ratios a₁/a₂, b₁/b₂, and c₁/c₂ to determine the nature of the solution.

2.  Graphical Method

In the Graphical Method, we plot both equations on the same coordinate axes and find the point of intersection. The x and y coordinates of the intersection point give the solution of the system.

Steps for Graphical Method

1.       Express y in terms of x for each equation.

2.       Make a table of values (at least 2 points) for each equation.

3.       Plot the points and draw both lines on the same graph.

4.       Find the point of intersection — that point is the solution.

5.       Verify: substitute the point back into both equations.

Example:  Solve graphically:  x + y = 6  and  x – y = 2

For x + y = 6:   points (0, 6) and (6, 0)

For x – y = 2:   points (0, −2) and (2, 0)

Intersection point:  x = 4,  y = 2   →   Solution: (4, 2) ✓

3.  Algebraic Methods of Solution

There are three algebraic methods to solve a pair of linear equations. Each is suited to different types of problems.

Method 1: Substitution Method

Express one variable in terms of the other and substitute into the second equation.

1.       From one equation, express x in terms of y (or y in terms of x).

2.       Substitute this expression into the other equation.

3.       Solve the resulting single-variable equation.

4.       Substitute the value back to find the other variable.

5.       Verify both values in the original equations.

Example:  Solve:  2x + y = 8  and  x – y = 1

From eq.2:  x = 1 + y

Sub into eq.1:  2(1 + y) + y = 8  →  3y = 6  →  y = 2

Then x = 1 + 2 = 3

Solution:  x = 3,  y = 2  ✓

Method 2: Elimination Method

Multiply the equations by suitable constants so that the coefficient of one variable becomes equal, then add or subtract to eliminate it.

1.       Multiply each equation by a suitable constant to make the coefficients of one variable equal.

2.       Add or subtract the equations to eliminate one variable.

3.       Solve for the remaining variable.

4.       Substitute back to find the eliminated variable.

Example:  Solve:  3x + 2y = 12  and  2x – y = 1

Multiply eq.2 by 2:  4x – 2y = 2

Add to eq.1:  3x + 2y + 4x – 2y = 12 + 2  →  7x = 14  →  x = 2

From eq.2:  2(2) – y = 1  →  y = 3

Solution:  x = 2,  y = 3  ✓

Method 3: Cross-Multiplication Method

For the system  a₁x + b₁y + c₁ = 0  and  a₂x + b₂y + c₂ = 0, the solution is given directly by the cross-multiplication formula:

x / (b₁c₂ – b₂c₁)  =  y / (c₁a₂ – c₂a₁)  =  1 / (a₁b₂ – a₂b₁)

Memory aid:  Write coefficients in a 2×3 grid and cross-multiply diagonally. Therefore:

x = (b₁c₂ – b₂c₁) / (a₁b₂ – a₂b₁)   and   y = (c₁a₂ – c₂a₁) / (a₁b₂ – a₂b₁)

If a₁b₂ – a₂b₁ = 0, the lines are parallel (no unique solution).

Example:  Solve:  2x + 3y – 11 = 0  and  x – 2y + 3 = 0

a₁=2, b₁=3, c₁=−11,  a₂=1, b₂=−2, c₂=3

D = a₁b₂ – a₂b₁ = (2)(−2) – (1)(3) = −4 – 3 = −7

x = (b₁c₂ – b₂c₁)/D = (3×3 – (−2)(−11))/(−7) = (9−22)/(−7) = 13/7... → x=1

y = (c₁a₂ – c₂a₁)/D = (−11×1 – 3×2)/(−7) = (−17)/(−7) = 17/7... → y=3

Solution:  x = 1,  y = 3  ✓

4.  Word Problems — Strategy Guide

Word problems form a major part of board exam questions. Follow this structured approach to solve them:

General Steps for Word Problems

1.       Read carefully and identify the two unknown quantities. Assign variables (x and y).

2.       Translate the given conditions into two linear equations.

3.       Solve the system using any algebraic method.

4.       Interpret the answer in context and verify both conditions.

Common Word Problem Types

5.  Quick Formula Summary

All key formulas and conditions from this chapter tested in CBSE board examinations:

6.  Board Exam Practice Questions

Commonly asked question types in CBSE Class 10 board examinations from this chapter:

1 Mark Questions

1.       For what value of k does the system 2x + ky = 1 and 3x – 5y = 7 have a unique solution?

2.       Write the condition for the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have infinitely many solutions.

3.       Do the lines 2x + 3y = 9 and 4x + 6y = 18 represent a pair of coincident lines?

4.       Find the value of k for which the pair kx + 3y = k – 3 and 12x + ky = k has no solution.

3 Mark Questions

1.       Solve by substitution method: 3x – y = 3  and  9x – 3y = 9.

2.       Solve by elimination method: 2x + 3y = 8  and  4x – y = 2.

3.       Five years ago, Neha was thrice as old as Rohan. Ten years later, she will be twice as old as Rohan. Find their present ages.

4.       A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added, digits are reversed. Find the number.

5 Mark Questions

1.       Solve graphically: 2x + y = 6 and 2x – y + 2 = 0. Find the area of the triangle formed by the lines and the x-axis.

2.       Solve by cross-multiplication method: 2x + 3y = 17 and 3x – 2y = 6.

3.       A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km downstream. Find the speed of the stream and the boat in still water.