Pair of Linear Equations in Two Variables

For the 2025–2026 academic year, the most important topics in Class 10 Maths for the chapter on

Pair of Linear Equations in Two Variables focus on the nature of solutions and the algebraic methods for solving them. The graphical method remains important for understanding the different types of solutions, while the cross-multiplication method is not included in the syllabus. 

Key topics in linear equations 

Consistent and inconsistent systems 

You must be able to compare the ratios of the coefficients (

a1/a2a sub 1 / a sub 2𝑎1/𝑎2,

b1/b2b sub 1 / b sub 2𝑏1/𝑏2, and

c1/c2c sub 1 / c sub 2𝑐1/𝑐2) to determine the nature of the solutions without solving the equations. 

• Intersecting lines (Unique solution): The pair of equations is consistent, and the lines will intersect at a single point.
  • Condition: a1/a2≠b1/b2a sub 1 / a sub 2 is not equal to b sub 1 / b sub 2𝑎1/𝑎2≠𝑏1/𝑏2.
• Coincident lines (Infinitely many solutions): The pair of equations is dependent and consistent. The lines overlap entirely, meaning every point on the line is a solution.
  • Condition: a1/a2=b1/b2=c1/c2a sub 1 / a sub 2 equals b sub 1 / b sub 2 equals c sub 1 / c sub 2𝑎1/𝑎2=𝑏1/𝑏2=𝑐1/𝑐2.
• Parallel lines (No solution): The pair of equations is inconsistent. The lines never intersect, so there is no solution.
  • Condition: a1/a2=b1/b2≠c1/c2a sub 1 / a sub 2 equals b sub 1 / b sub 2 is not equal to c sub 1 / c sub 2𝑎1/𝑎2=𝑏1/𝑏2≠𝑐1/𝑐2. 

Algebraic methods for solving equations 

Mastery of the two algebraic methods is essential for both simple equations and word problems. 

• Substitution Method: Isolate a variable in one equation and substitute its expression into the other equation. This method is often easier when a variable has a coefficient of 1 or -1.
• Elimination Method: Multiply one or both equations by suitable constants to make the coefficients of one variable the same (or opposites), then add or subtract the equations to eliminate that variable. 

Word problems (situational problems) 

Translating word problems into a pair of linear equations is a critical skill, as these questions often carry high marks. 

• Common types: Practice problems based on ages, fractions, fixed charges (like taxi fares or library fees), and speed, distance, and time (especially boat problems involving upstream and downstream travel).
• Reducible equations: You may encounter equations that are not initially linear but can be reduced to a pair of linear equations by making a substitution (e.g., let 1/x=p1 / x equals p1/𝑥=𝑝 and 1/y=q1 / y equals q1/𝑦=𝑞). 

Deleted topics for 2025–2026 

• Cross-Multiplication Method: This algebraic method has been explicitly removed from the syllabus for the 2025–2026 academic session.
• Graphical Solution: While graphical representation is crucial for conceptual understanding, and questions on identifying consistent/inconsistent systems graphically may appear, questions requiring you to solve a system by plotting graphs may be excluded. However, plotting graphs to find vertices of a triangle with the axes is still relevant