Chapter Hub
Pair of Linear Equations in Two Variables
This chapter teaches how to represent and solve two linear equations in two variables using graphs and algebraic methods. Students learn when a pair has one solution, no solution, or infinitely many solutions, and how to read that from the equations and the graph. It is an important Class 10 chapter for exam writing because questions often test conditions for consistency, solution checking, substitution, elimination, cross-multiplication, and word problems based on age, numbers, and daily-life situations.
Difficulty
Medium
Study time
96-120 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 96 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
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Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
Pair of linear equations in two variables
A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.
Graphical representation of a pair of linear equations
Graphical representation means drawing both linear equations on the same Cartesian plane and using their position to understand the solution.
Unique solution and intersecting lines
A pair of linear equations has a unique solution when the two lines intersect at exactly one point.
No solution and parallel lines
A pair of linear equations has no solution when the two lines are parallel and never meet.
Infinitely many solutions and coincident lines
A pair of linear equations has infinitely many solutions when both equations represent the same line.
Algebraic solution by substitution
The substitution method solves one equation for one variable and substitutes that expression into the other equation.
Algebraic solution by elimination
The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.
Algebraic solution by cross-multiplication
Cross-multiplication is a method used to solve a pair of linear equations written in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
Reducing word problems to linear equations
Reducing a word problem means converting the given statements into two linear equations in two variables.
Condition for consistency using ratios
The ratio condition tells whether a pair of linear equations has one solution, no solution, or infinitely many solutions.
Checking obtained solution in both equations
Checking means substituting the obtained values of x and y into both equations to confirm that the pair is correct.
Age and number based linear models
Age and number models are word problems where unknown ages or numbers are represented by variables and converted into linear equations.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Pair of linear equations in two variables
- Graphical representation of a pair of linear equations
- Unique solution and intersecting lines
- No solution and parallel lines
- Infinitely many solutions and coincident lines
- Algebraic solution by substitution
- Algebraic solution by elimination
- Algebraic solution by cross-multiplication
Common Traps
- Stopping after finding only one variable instead of finding the ordered pair.
- Comparing only two ratios and forgetting the constant term ratio.
- Adding or subtracting equations with the wrong sign choice in elimination.
- Ignoring the sign of c in cross-multiplication.
- Checking the answer in only one equation and not both.
- Mixing up sum, difference, and 'more than' statements in word problems.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, or conditions
- Short answer: worked method, example, or reason-based explanation
- Case-based: chapter scenario with concept-linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Two lines can meet once, never meet, or overlap completely.
- Graphical reading gives the nature of solutions quickly.
- Substitution works best when one variable is already isolated.
- Elimination works best when coefficients can be matched or are already opposite.
- Cross-multiplication is powerful only when standard form is written neatly.
- Word problems become easy after choosing variables and writing two clear equations.
- Checking the final pair protects marks from small sign or arithmetic mistakes.
- Pair of linear equations in two variables: A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, wi…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
Try a 15-question quiz from this chapter. Get instant score and unlock concept-wise analytics.
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