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
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Meaning and representation
Covers Pair of linear equations in two variables and Graphical representation of a pair of linear equations.
Open mapNature of solutions
Covers Unique solution and intersecting lines, No solution and parallel lines and Infinitely many solutions and coincident lines.
Open mapAlgebraic methods
Covers Algebraic solution by substitution, Algebraic solution by elimination and Algebraic solution by cross-multiplication.
Open mapApplications and checks
Covers Reducing word problems to linear equations, Condition for consistency using ratios, Checking obtained solution in both equations and Age and number based linear models.
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Meaning and representation
Covers Pair of linear equations in two variables and Graphical representation of a pair of linear equations.
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.
Nature of solutions
Covers Unique solution and intersecting lines, No solution and parallel lines and Infinitely many solutions and coincident lines.
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 methods
Covers Algebraic solution by substitution, Algebraic solution by elimination and Algebraic solution by cross-multiplication.
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.
Applications and checks
Covers Reducing word problems to linear equations, Condition for consistency using ratios, Checking obtained solution in both equations and Age and number based linear models.
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
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
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.
How CBSE Asks This Chapter
- Pair of linear equations in two variables
The exam may ask you to classify a pair, solve it by a method, or identify the ordered pair that satisfies both equations.
- Graphical representation of a pair of linear equations
The exam may ask you to draw the graph, identify the type of solution, or read the solution from the intersection point.
- Unique solution and intersecting lines
The exam may ask for the nature of the solution, the graph type, or the algebraic condition for a unique solution.
- No solution and parallel lines
The exam may ask for the nature of consistency, the ratio condition, or the graph of a pair with no solution.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Pair of linear equations in two variables
Remember
A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero.Trap to avoid
Many students treat the two equations separately and write two different answers.
Graphical representation of a pair of linear equations
Remember
For a line, at least two points are enough to draw its graph accurately, and the solution of a pair is read from the relation between the two lines.Trap to avoid
Students often draw rough lines without using enough scale or without marking intercepts correctly, which gives the wrong solution type.
Unique solution and intersecting lines
Remember
For one unique solution, the two lines must not be parallel and must not coincide.Trap to avoid
Students may confuse intersecting lines with parallel lines, especially when the graph is drawn approximately.
No solution and parallel lines
Remember
For no solution, a1/a2 = b1/b2, but a1/a2 is not equal to c1/c2.Trap to avoid
Students often think parallel lines can meet after extension, but in Euclidean geometry they never intersect.
Infinitely many solutions and coincident lines
Remember
For infinitely many solutions, a1/a2 = b1/b2 = c1/c2.Trap to avoid
Students may think infinitely many solutions means the answer is uncertain.
Algebraic solution by substitution
Remember
If x = expression or y = expression in one equation, substitute that expression into the other equation.Trap to avoid
Students often substitute into the same equation they used to isolate the variable, which gives no new information.
Algebraic solution by elimination
Remember
Make the coefficients of one variable equal if needed, then add or subtract the equations to eliminate that variable.Trap to avoid
Students often add when subtraction is needed, or subtract when addition is needed, so the intended variable does not cancel.
Algebraic solution by cross-multiplication
Remember
If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then x = (b1c2 - b2c1)/(a1b2 - a2b1) and y = (c1a2 - c2a1)/(a1b2 - a2b1).Trap to avoid
Students often place the coefficients in the wrong positions in the formula or miss the minus signs in the numerators.
Reducing word problems to linear equations
Remember
Use two unknowns for two conditions, and translate keywords like sum, difference, total, more than, and less than into equations carefully.Trap to avoid
Students sometimes choose the wrong variables or translate the same statement twice instead of forming two different equations.
Condition for consistency using ratios
Remember
One solution: a1/a2 not equal to b1/b2.Trap to avoid
Students often compare only one ratio and ignore the constant term ratio, which leads to a wrong conclusion.
Checking obtained solution in both equations
Remember
Substitute the obtained x and y in each original equation and verify that both left and right sides match.Trap to avoid
Students often check the answer in only one equation and assume the pair is correct.
Age and number based linear models
Remember
Present age, past age, and future age are related by simple addition or subtraction of years.Trap to avoid
Students often use the same variable for both people or forget to express future and past ages correctly.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Pair of linear equations in two variables | A linear equation in two variables is usually written as ax + by + c = 0, where a and b are not both zero. | Many students treat the two equations separately and write two different answers. |
| Graphical representation of a pair of linear equations | For a line, at least two points are enough to draw its graph accurately, and the solution of a pair is read from the relation between the two lines. | Students often draw rough lines without using enough scale or without marking intercepts correctly, which gives the wrong solution type. |
| Unique solution and intersecting lines | For one unique solution, the two lines must not be parallel and must not coincide. | Students may confuse intersecting lines with parallel lines, especially when the graph is drawn approximately. |
| No solution and parallel lines | For no solution, a1/a2 = b1/b2, but a1/a2 is not equal to c1/c2. | Students often think parallel lines can meet after extension, but in Euclidean geometry they never intersect. |
| Infinitely many solutions and coincident lines | For infinitely many solutions, a1/a2 = b1/b2 = c1/c2. | Students may think infinitely many solutions means the answer is uncertain. |
| Algebraic solution by substitution | If x = expression or y = expression in one equation, substitute that expression into the other equation. | Students often substitute into the same equation they used to isolate the variable, which gives no new information. |
| Algebraic solution by elimination | Make the coefficients of one variable equal if needed, then add or subtract the equations to eliminate that variable. | Students often add when subtraction is needed, or subtract when addition is needed, so the intended variable does not cancel. |
| Algebraic solution by cross-multiplication | If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then x = (b1c2 - b2c1)/(a1b2 - a2b1) and y = (c1a2 - c2a1)/(a1b2 - a2b1). | Students often place the coefficients in the wrong positions in the formula or miss the minus signs in the numerators. |
| Reducing word problems to linear equations | Use two unknowns for two conditions, and translate keywords like sum, difference, total, more than, and less than into equations carefully. | Students sometimes choose the wrong variables or translate the same statement twice instead of forming two different equations. |
| Condition for consistency using ratios | One solution: a1/a2 not equal to b1/b2. | Students often compare only one ratio and ignore the constant term ratio, which leads to a wrong conclusion. |
| Checking obtained solution in both equations | Substitute the obtained x and y in each original equation and verify that both left and right sides match. | Students often check the answer in only one equation and assume the pair is correct. |
| Age and number based linear models | Present age, past age, and future age are related by simple addition or subtraction of years. | Students often use the same variable for both people or forget to express future and past ages correctly. |
Practice
Use short concept checks first, then move into the full chapter test.
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