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Class X Mathematics
Chapter 3: Pair of Linear Equations in Two Variables
Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.
Syllabus Sections
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Chapter Overview
Welcome to Class X Mathematics: Pair of Linear Equations in Two Variables. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.
Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.
Prerequisite Concepts
Linear Equations in Two Variables
About This Chapter
This comprehensive study guide for Pair of Linear Equations in Two Variables is designed for Class X students following the CBSE and NCERT Mathematics curriculum. It covers 6 key subtopics including Graphical method of solution, Consistency conditions (Consistent, Inconsistent, Coincident), Substitution method, and 3 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.
The chapter includes 2 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.
Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Pair of Linear Equations in Two Variables, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.
What You'll Learn in This Chapter
By the end of studying Pair of Linear Equations in Two Variables for Class X, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:
Determine if a system of two equations has 0, 1, or infinite solutions.
Solve linear systems algebraically using substitution and elimination.
Formulate and solve speed-distance and work-rate word problems.
Prerequisites for This Chapter
Before studying Pair of Linear Equations in Two Variables, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:
Linear Equations in Two Variables
If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Pair of Linear Equations in Two Variables with full confidence.
Real-World Applications of Pair of Linear Equations in Two Variables
Students often wonder “Where will I use Pair of Linear Equations in Two Variablesin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:
Budget Planning & Finance
Balancing income and expenses across categories can be modeled as systems of linear equations to find optimal allocation.
Supply and Demand Economics
Economists model market equilibrium by finding the intersection point of linear supply and demand curves.
Mixture Problems in Chemistry
Determining concentrations when mixing solutions requires solving pairs of linear equations simultaneously.
Traffic Flow Optimization
City planners use systems of equations to model vehicle flow at intersections and optimize traffic signal timing.
Understanding the real-world relevance of Pair of Linear Equations in Two Variables not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.
Study Tips for Pair of Linear Equations in Two Variables
Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Pair of Linear Equations in Two Variables:
✏️
Practice Step-by-Step
Write out every intermediate step when solving problems. Skipping steps is the most common source of errors in calculation-heavy chapters. Build speed only after achieving consistent accuracy.
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Verify by Back-Substitution
After finding your answer, substitute it back into the original equation to verify correctness. This simple habit catches most arithmetic and sign errors before they cost you marks.
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Maintain an Error Log
Keep a dedicated notebook of mistakes you make during practice. Review it weekly to identify patterns — you will notice the same types of errors recurring and can actively work to eliminate them.
Pro Tip: Consistency beats intensity. Studying Pair of Linear Equations in Two Variables for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.
Detailed Topic Breakdown
Detailed Subtopics Study Guide
Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:
1Graphical method of solution
Concept Explanation
Solving a system of two linear equations graphically involves plotting both lines on the same coordinate grid. The coordinates of the intersection point (x, y) represent the unique solution to the system.
Mathematical Representation
L_1 \cap L_2 = \{(x_0, y_0)\}
Study Guideline: Find two or three points for each equation, draw the lines, and locate their intersection. Verify the intersection point mathematically by plugging it back into both equations.
2Consistency conditions (Consistent, Inconsistent, Coincident)
Concept Explanation
Consistency conditions determine the number of solutions for a pair of linear equations. A system is consistent (unique solution if lines intersect, infinite solutions if lines coincide) or inconsistent (no solution if lines are parallel).
Mathematical Representation
\text{Unique: } \frac{a_1}{a_2} \neq \frac{b_1}{b_2}, \quad \text{Infinite: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, \quad \text{No Solution: } \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}
Study Guideline: Compare the ratio of the coefficients of x, y, and the constant terms to determine the nature of solutions without drawing the graph.
3Substitution method
Concept Explanation
The substitution method solves a system of linear equations by expressing one variable in terms of the other from one equation, and then substituting this expression into the second equation to get a single-variable equation.
Mathematical Representation
x = \frac{d - by}{a} \implies \text{Substitute into } a_2x + b_2y + c_2 = 0
Study Guideline: Choose the equation and variable that are easiest to isolate (e.g., a variable with a coefficient of 1) to avoid fractions early in the calculation.
4Elimination method
Concept Explanation
The elimination method solves a system of linear equations by multiplying one or both equations by suitable non-zero constants so that the coefficients of one variable become equal (or opposite). Adding or subtracting the equations then eliminates that variable.
Mathematical Representation
a_1 x + b_1 y = c_1 \, [\times a_2] \, \text{ and } \, a_2 x + b_2 y = c_2 \, [\times a_1] \implies \text{Subtract to eliminate } x
Study Guideline: Multiply the equations so that coefficients of either x or y match. Watch out for signs when subtracting the equations.
5Cross-multiplication method overview
Concept Explanation
The cross-multiplication method is a formula-based algebraic method to solve a pair of linear equations in standard form. It calculates x and y directly using ratios of coefficients.
Mathematical Representation
\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}
Study Guideline: Write the coefficients in the pattern: b, c, a, b. Cross-multiply down-wards minus cross-multiplying up-wards to find the denominators.
6Equations reducible to linear form
Concept Explanation
Equations with variables in denominators can be simplified to linear equations using cross-multiplication.
Mathematical Representation
\frac{ax+b}{cx+d} = k \implies ax+b = k(cx+d)
Study Guideline: Multiply the entire equation by the common denominator to clear out fractions.
Learning Outcomes
By the end of this chapter, students will be able to master and solve questions on these outcomes:
Determine if a system of two equations has 0, 1, or infinite solutions.
Solve linear systems algebraically using substitution and elimination.
Formulate and solve speed-distance and work-rate word problems.
Sub-Topics Directory
Graphical method of solutionConsistency conditions (Consistent, Inconsistent, Coincident)Substitution methodElimination methodCross-multiplication method overviewEquations reducible to linear form
Key Formulas & Equations
Click Copy to duplicateSystem Consistency condition
\frac{a_1}{a_2} \neq \frac{b_1}{b_2}
Condition for a unique solution (intersecting lines).
Infinite Solutions condition
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
Condition for coincident lines.
Related Formulas from Adjacent Chapters
Quadratic Roots Sum
\alpha + \beta = -\frac{b}{a}
[From Polynomials] Sum of roots of ax² + bx + c.
Quadratic Roots Product
\alpha \beta = \frac{c}{a}
[From Polynomials] Product of roots of ax² + bx + c.
The Quadratic Formula
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
[From Quadratic Equations] Finds the exact roots of ax² + bx + c = 0.
Interactive Problem Solver
**Solving by Elimination**
Solve the linear system: x + y = 5 and 2x - 3y = 4.
Scientific Calculator Active: A custom solver for the specific subtopics of this chapter is currently in development. In the meantime, please use our comprehensive Scientific Functions Calculator below to verify your study equations.
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Adjust Inputs
45
Calculated Results
Function Output
0.71
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Deterministic Mathematical Simulation Engine • Verified Calculations
Date: 9/1/2026
Time: 9:29:40 AM
Scientific Functions Calculator
Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.
1. Input Parameters
| Parameter | Value | Unit |
|---|---|---|
| Value (X) | 45 | — |
| Function | sin | — |
2. Calculated Results
| Metric | Calculated Output |
|---|---|
| Function Output | 0.707 |
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Real-time calculationsStep-by-step solving
1. Collect input parameters:
- operand1 = 45
- operation = sin
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3. Compile calculated results:
- calcResult = 0.7071
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Class X CBSE Board Practice Questions
Review and solve 52 real Board paper questions from the 20 mock sample sets matching Pair of Linear Equations in Two Variables.
Filter by Section:
Sample Paper 01 | Question 6Section A (1 Mark)
The pair of linear equations 2x = 5y + 6 and 15y = 6x - 18 represents two lines which are:
Sample Paper 01 | Question 20Section A (1 Mark)
Assertion (A): If the co-ordinates of the mid-points of the sides AB and AC of ΔABC are D(3,5) and E(-3,-3) respectively, then BC = 20 units Reason (R): The line joining the mid points of two sides of a triangle is parallel to the third side and equal to half of it.
Sample Paper 01 | Question 23Section B (2 Marks)
Solve the following system of linear equations: 7x - 2y = 5 and 8x + 7y= 15 and verify your
Sample Paper 01 | Question 34Section D (5 Marks)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Sample Paper 02 | Question 23Section B (2 Marks)
For what value of k will the following system of linear equations have no solution? 3x + y = 1; (2k - 1) x + (k - 1) y = 2k + 1
Sample Paper 02 | Question 30Section C (3 Marks)
Solve the following linear equations: 152x - 378y = -74 and -378x + 152y = -604
Sample Paper 03 | Question 21Section B (2 Marks)
For what value of k for which the following pair of linear equations have infinitely many solutions: 2x + 3y = 7, (k - 1)x + (k + 2)y = 3k is
Sample Paper 03 | Question 23Section B (2 Marks)
X is a point on the side BC of ∆ABC. XM and XN are drawn parallel to AB and AC respectively meeting AB in N and AC in M. MN produced meets CB produced at T. Prove that TX 2 = TB × TC.
Sample Paper 03 | Question 37Section E (4 Marks)
Case Study - 2 In a GPS, The lines that run east -west are known as lines of latitude, and the lines running north - south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below. Based on the above information
Sample Paper 04 | Question 6Section A (1 Mark)
The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is
Sample Paper 04 | Question 23Section B (2 Marks)
For what values of k will the following pair of linear equations have infinitely many solutions? kx + 3y - (k - 3) = 0 and 12x + ky - k = 0
Sample Paper 04 | Question 29Section C (3 Marks)
In the below figure, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠AOB = 90°. In ∆AO'D and ∆AOC, ∠ADO' = ∠ACO = 90° ∠A = ∠A (Common) ∴ ∆AO'D ~ ∆AOC (By AA similarity) ⇒ ' 'AO DO AO CO= Now, AO = AO' + O' X + XO = 3r ' 1 3 3 DO r CO r\ = =
Sample Paper 04 | Question 34Section D (5 Marks)
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. In the figure, find EC if AD/DB = AE/EC using the above theorem.
Sample Paper 05 | Question 21Section B (2 Marks)
For what value of k for which the following pair of linear equations have inf initely many solutions: 2x + 3y = 7, (k - 1)x + (k + 2)y = 3k is
Sample Paper 05 | Question 28Section C (3 Marks)
Prove that a parallelogram circumscribing a circle is a rhombus
Sample Paper 06 | Question 5Section A (1 Mark)
The vertices of a parallelogram in order are A(1, 2), B(4, y), C(x, 6) and D(3, 5). Then (x, y) is:
Sample Paper 06 | Question 35Section D (5 Marks)
Prove that “If a line is drawn parallel to one side of a triangle to intersect the other two sides i n distinct points, the other two sides are divided in the same ratio.” In the figure, find EC if AD/DB = AE/EC using the above theorem.
Sample Paper 06 | Question 36Section E (4 Marks)
Case Study - 1 In a GPS, The lines that run east -west are known as lines of latitude, and the lines running north-south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below. Based on the above information
Sample Paper 07 | Question 35Section D (5 Marks)
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.
Sample Paper 08 | Question 7Section A (1 Mark)
If the lines 3x + 2ky - 2 = 0 and 2x + 5y + 1 = 0 are parallel, then what is the value of k?
Sample Paper 08 | Question 10Section A (1 Mark)
If the lines 3x + 2ky - 2 = 0 and 2x + 5y + 1 = 0 are parallel, then what is the value of k?
Sample Paper 08 | Question 19Section A (1 Mark)
Assertion (A): In ∆ABC, DE || BC such that AD = (7x - 4) cm, AE = (5x - 2) cm, DB = (3x + 4) cm and EC = 3x cm than x equal to 5. Reason (R): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distant point, than the other two sides are divided in the same ratio.
Sample Paper 08 | Question 25Section B (2 Marks)
If the system of equations 2x + 3y = 7 and (a + b)x + (2a - b)y = 21 has infinitely many solutions, then find a and b.
Sample Paper 08 | Question 35Section D (5 Marks)
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.
Sample Paper 09 | Question 27Section C (3 Marks)
Prove that a parallelogram circumscribing a circle is a rhombus
Sample Paper 09 | Question 32Section D (5 Marks)
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.
Sample Paper 10 | Question 11Section A (1 Mark)
If the vertices of a parallelogram PQRS taken in order are P(3, 4), Q(-2, 3) and R(-3, -2), then the coordinates of its fourth vertex S are
Sample Paper 10 | Question 18Section A (1 Mark)
If the system of equations 3x + y =1 and (2k - 1)x + (k - 1)y = 2k + 1 is inconsistent, then k =
Sample Paper 10 | Question 19Section A (1 Mark)
Assertion (A): The tangents drawn at the end points of a diameter of a circle, are parallel. Reason (R): Diameter of a circle is the longest chord.
Sample Paper 10 | Question 33Section D (5 Marks)
(a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Sample Paper 11 | Question 13Section A (1 Mark)
The point of intersection of the line represented by 3x - y = 3 and y-axis is given by
Sample Paper 13 | Question 1Section A (1 Mark)
The value of k for which the pair of equation kx - y = 2 and 6x - 2y = 3 has unique solution
Sample Paper 13 | Question 21Section B (2 Marks)
Find the value of m for which the pair of linear equations: 2x + 3y - 7 = 0 and (m - 1) x + (m + 1) y = (3m - 1) has infinitely many solutions
Sample Paper 13 | Question 35Section D (5 Marks)
Prove that “If a line is drawn parallel to one side of a triangle to intersect the other two sides i n distinct points, the other two sides are divided in the same ratio.” In the figure, find EC if AD/DB = AE/EC using the above theorem.
Sample Paper 14 | Question 16Section A (1 Mark)
Find the value of k so that the following system of equations has no solution: 3x - y - 5 = 0, 6x - 2y + k = 0
Sample Paper 14 | Question 25Section B (2 Marks)
Determine the values of a and b for which the following system of linear equations has infinite solutions: 2x - (a - 4) y = 2b + 1; 4x - (a - 1) y = 5b - 1
Sample Paper 14 | Question 32Section D (5 Marks)
If a line is drawn parallel to one side of a triangle, prove that the other two sides are divided in the same ratio. Using this theorem, find x in below figure, if MN || QR, PM = x cm, MQ = 10 cm, PN = (x - 2) cm, NR = 6 cm
Sample Paper 15 | Question 3Section A (1 Mark)
Solve the system of equations: x + y = 6 and x - y = 4.
Sample Paper 15 | Question 31Section C (3 Marks)
A student room rent/hostel charges consist of fixed charges and daily food cost. When a student A takes food for 22 days, he pays Rs 1380. Student B pays Rs 1680 for 28 days. Find fixed charges and daily food cost.
Sample Paper 16 | Question 2Section A (1 Mark)
The pair of equations x + 2y + 5 = 0 and -3x - 6y + 1 = 0 has:
Sample Paper 16 | Question 27Section C (3 Marks)
Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in total. Find how many notes of Rs 50 and Rs 100 she received.
Sample Paper 16 | Question 31Section C (3 Marks)
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Sample Paper 17 | Question 4Section A (1 Mark)
The pair of linear equations x + 2y + 5 = 0 and - 3x = 6y - 1 has
Sample Paper 17 | Question 29Section C (3 Marks)
Solve the following system of linear equations graphically: x - y + 1 = 0 and x + y = 5
Sample Paper 17 | Question 34Section D (5 Marks)
Prove that if a line is a drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Using the above theorem. Prove that = if LM || CB and LN || CD as shown in the figure.
Sample Paper 17 | Question 35Section D (5 Marks)
Solve the following system of linear equations graphically: 2x + 3y = 12 and x - y = 1. Find the coordinates of the vertices of the triangle formed by these lines and the y-axis.
Sample Paper 18 | Question 19Section A (1 Mark)
Assertion (A): The pair of linear equations 5x + 2y + 6 = 0 and 7x + 6y + 18 = 0 have infinitely many solutions. Reason (R): The pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 have infinitely many solutions, if 1 1 1 2 2 2 a b c a b c= =
Sample Paper 19 | Question 21Section B (2 Marks)
Find the value of p if the pair of equations 2x + 3y - 5 = 0 and px - 6y - 8 = 0 has a unique solution.
Sample Paper 20 | Question 7Section A (1 Mark)
The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is
Sample Paper 20 | Question 16Section A (1 Mark)
Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABC
Sample Paper 20 | Question 26Section C (3 Marks)
In the below figure, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠AOB = 90°.
Sample Paper 20 | Question 33Section D (5 Marks)
Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.
Critical Mistakes to Avoid
Recognizing and eliminating common errors is a key step to scoring high marks in school and board exams:
!Incorrect Discriminant Signs
Evaluating D = b² - 4ac incorrectly when a or c is negative, resulting in wrong sign operations (e.g. -4 × 2 × -3 should add +24, not subtract 24).
!Forgetting the ± Sign
When solving equations like (x-3)² = 16, stating only x-3 = 4, and missing the negative root x-3 = -4.
!Dividing by Variables
Simplifying x² = 3x by dividing both sides by x, which removes the root x = 0. Always factorize instead: x(x - 3) = 0.
Revision Summary Cards
1Standard Form
Quadratic equations must always be written as ax² + bx + c = 0 before applying coefficients.
2The Quadratic Formula
Roots are computed as x = [-b ± √(b² - 4ac)] / 2a.
3Discriminant Rule
D = b² - 4ac determines roots. D > 0 means 2 real roots, D = 0 is 1 real root, and D < 0 yields complex conjugate roots.
4Prerequisites Checked
Ensure all prerequisite concepts are fully understood before working on advanced calculations.
5Step-by-Step Checking
Always check equations step-by-step to avoid simple sign and arithmetic errors.
Frequently Asked Questions
No. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n complex roots. A quadratic equation is a second-degree polynomial (n = 2) and thus can have at most two roots.
When the discriminant D = b² - 4ac is negative, the square root term contains a negative value, yielding roots with the imaginary unit i (where i = √-1). These are complex numbers representing points where the parabola does not cross the x-axis.
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Worksheet Details:
- • Title: Class X Math: Pair of Linear Equations in Two Variables Printable Sheet
- • Format: Print Optimized CSS stylesheet
- • Sections: Formulas checklist, 5 assessment questions, worked keys
NexPro Education Worksheet
Class X Mathematics: Pair of Linear Equations in Two Variables
Name: ______________________Date: __________________Score: ______ / 5
Section I: Key Formulas
System Consistency condition
\frac{a_1}{a_2} \neq \frac{b_1}{b_2}
Infinite Solutions condition
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
Section II: Answer the Questions
1. Solve the following system of linear equations: 7x - 2y = 5 and 8x + 7y= 15 and verify your
Solve and write the step-by-step solution in the workspace below.
2. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Solve and write the step-by-step solution in the workspace below.
3. For what value of k will the following system of linear equations have no solution? 3x + y = 1; (2k - 1) x + (k - 1) y = 2k + 1
Solve and write the step-by-step solution in the workspace below.
4. Solve the following linear equations: 152x - 378y = -74 and -378x + 152y = -604
Solve and write the step-by-step solution in the workspace below.
5. For what value of k for which the following pair of linear equations have infinitely many solutions: 2x + 3y = 7, (k - 1)x + (k + 2)y = 3k is
Solve and write the step-by-step solution in the workspace below.