CBSE Class 10 Mathematics (2026–27)
Chapter 3: Pair of Linear Equations in Two Variables
20 Important Questions and Answers
This chapter covers graphical representation, consistency of equations, algebraic methods (substitution and elimination), and simple situational problems. These topics are prescribed in the CBSE Class 10 Mathematics syllabus for 2026–27.
Q1. What is a pair of linear equations in two variables?
Answer:
A pair of linear equations in two variables consists of two equations involving the same two variables, generally represented as:
(a_1x+b_1y+c_1=0) and (a_2x+b_2y+c_2=0).
The values of (x) and (y) that satisfy both equations simultaneously form the solution of the pair. Geometrically, each equation represents a straight line on a graph. The solution depends on the position of the lines. If the lines intersect, there is one unique solution. If they are parallel, there is no solution. If they coincide, there are infinitely many solutions. Pair of linear equations are widely used to solve real-life problems involving age, money, distance, and mixtures.
Q2. Explain the graphical method of solving a pair of linear equations.
Answer:
In the graphical method, both equations are represented as straight lines on the Cartesian plane. First, two or more points satisfying each equation are found. These points are plotted and joined to obtain the graph of each line. The point where the two lines intersect gives the solution of the pair of equations. If the lines meet at exactly one point, there is a unique solution. If the lines are parallel, they never meet and therefore have no solution. If both lines overlap completely, they represent the same line and have infinitely many solutions. The graphical method provides a visual understanding of the nature of solutions.
Q3. What do consistent and inconsistent systems mean?
Answer:
A pair of linear equations is called consistent if it has at least one solution. Consistent systems can be of two types: independent and dependent. An independent system has a unique solution because the lines intersect at one point. A dependent system has infinitely many solutions because the lines coincide. A pair of equations is called inconsistent when it has no solution. This happens when the two lines are parallel and never meet. The concept of consistency helps determine whether a given pair of equations can be solved simultaneously. Understanding consistency is important for analyzing the relationship between two linear equations.
Q4. State the conditions for a unique solution of a pair of linear equations.
Answer:
For the equations
(a_1x+b_1y+c_1=0) and (a_2x+b_2y+c_2=0),
a unique solution exists when:
[\frac{a_1}{a_2}\ne\frac{b_1}{b_2}]
This condition indicates that the two lines have different slopes and intersect at exactly one point. Since they meet at a single point, there is only one pair of values of (x) and (y) that satisfies both equations simultaneously. Such a pair of equations is called a consistent and independent system. In graphical representation, the two lines cross each other at one point, which represents the unique solution of the system.
Q5. What are the conditions for infinitely many solutions?
Answer:
A pair of linear equations has infinitely many solutions when:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}]
Under this condition, both equations represent the same straight line. Every point on the line satisfies both equations simultaneously. Therefore, there are unlimited pairs of values of (x) and (y) that satisfy the system. Such equations are called dependent equations and form a consistent system. Graphically, the two lines completely overlap each other, making them indistinguishable. This condition is important for identifying whether a pair of equations represents the same mathematical relationship.
Q6. What are the conditions for no solution?
Answer:
A pair of linear equations has no solution when:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}]
This condition means that the coefficients of (x) and (y) are proportional, but the constant terms are not. As a result, the two equations represent parallel lines. Since parallel lines never intersect, there is no common point satisfying both equations simultaneously. Therefore, no solution exists. Such a system is called inconsistent. Graphically, the lines remain at a fixed distance from each other and never meet. Recognizing this condition helps students quickly determine the nature of solutions without solving the equations completely.
Q7. Explain the substitution method.
Answer:
The substitution method is an algebraic technique used to solve a pair of linear equations. In this method, one equation is used to express one variable in terms of the other. This expression is then substituted into the second equation. As a result, the second equation contains only one variable, which can be solved easily. After finding the value of one variable, it is substituted back into either original equation to determine the other variable. This method is especially useful when one equation can be easily rearranged. The substitution method provides accurate solutions and is widely used in solving simultaneous linear equations.
Q8. Explain the elimination method.
Answer:
The elimination method is a systematic algebraic method used to solve a pair of linear equations. In this method, the coefficients of one variable are made equal by multiplying one or both equations by suitable numbers. The equations are then added or subtracted so that one variable gets eliminated. This leaves an equation in a single variable, which can be solved easily. After obtaining one variable, its value is substituted into either original equation to find the other variable. The elimination method is often preferred when coefficients can be made equal conveniently and provides quick and accurate solutions.
Q9. Solve: x + y = 7 and x – y = 1.
Answer:
Using the elimination method:
Given:
x + y = 7
x – y = 1
Adding both equations:
[(x+y)+(x-y)=7+1]
[2x=8]
[x=4]
Substituting x = 4 into x + y = 7:
[4+y=7]
[y=3]
Therefore, the solution is:
[(x,y)=(4,3)]
Verification:
4 + 3 = 7 and 4 – 3 = 1.
Hence, the pair (4,3) satisfies both equations. The graphical representation would show two intersecting lines meeting at the point (4,3), confirming a unique solution.
Q10. Solve: 2x + y = 11 and x + y = 7.
Answer:
Using elimination:
Given:
2x + y = 11
x + y = 7
Subtract the second equation from the first:
[(2x+y)-(x+y)=11-7]
[x=4]
Substitute x = 4 into x + y = 7:
[4+y=7]
[y=3]
Therefore, the solution is:
[(x,y)=(4,3)]
Verification:
2(4)+3=11 and 4+3=7.
Both equations are satisfied. Thus, the pair of equations has a unique solution represented by the point (4,3) on the graph where the two lines intersect.
Q11. Why are intersecting lines associated with a unique solution?
Answer:
When two straight lines intersect, they meet at exactly one point on the coordinate plane. This common point satisfies both equations simultaneously. Since there is only one such point, there is only one pair of values for (x) and (y) that fulfills both equations. Therefore, intersecting lines correspond to a unique solution. Algebraically, this occurs when the ratios of coefficients satisfy:
[\frac{a_1}{a_2}\ne\frac{b_1}{b_2}]
Such systems are called consistent and independent. Understanding this graphical interpretation helps students connect algebraic conditions with geometric representations and improves conceptual clarity.
Q12. Why do coincident lines have infinitely many solutions?
Answer:
Coincident lines are actually the same line represented by two equivalent equations. Every point on the line satisfies both equations simultaneously. Since a straight line contains infinitely many points, there are infinitely many solutions to the pair of equations. Algebraically, this happens when:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}]
The equations are dependent because one equation can be obtained by multiplying the other by a constant. Such systems are consistent because solutions exist, but the number of solutions is unlimited. Graphically, only one line appears because both equations overlap completely.
Q13. Why do parallel lines have no solution?
Answer:
Parallel lines never intersect regardless of how far they are extended. Since there is no common point between them, no pair of values of (x) and (y) can satisfy both equations simultaneously. Therefore, the system has no solution. Algebraically, parallel lines occur when:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}]
The equations have the same slope but different intercepts. Such a pair of equations is called inconsistent because no common solution exists. Graphically, the lines remain at a constant distance from each other and never meet.
Q14. What is meant by the solution of a pair of linear equations?
Answer:
The solution of a pair of linear equations is the ordered pair ((x,y)) that satisfies both equations simultaneously. When these values are substituted into both equations, each equation becomes true. Graphically, the solution corresponds to the point where the two lines intersect. If the lines intersect at one point, there is one solution. If they coincide, there are infinitely many solutions. If they are parallel, there is no solution. The concept of a solution is fundamental because it helps in solving practical problems involving unknown quantities represented by variables.
Q15. How are pair of linear equations used in daily life?
Answer:
Pair of linear equations are widely used to solve practical problems involving two unknown quantities. They help in determining the ages of people, prices of items, distances traveled, mixture compositions, and profit calculations. For example, if the total cost of pens and pencils is known along with another condition, two equations can be formed and solved to find individual prices. Similarly, problems involving speed and time can be converted into linear equations. These applications show how mathematical models represent real-life situations and help in finding accurate solutions. Therefore, pair of linear equations have significant practical importance.
Q16. Differentiate between graphical and algebraic methods.
Answer:
The graphical method involves plotting the equations on a coordinate plane and identifying the intersection point. It provides a visual understanding of the solution and the nature of the lines. However, exact values may not always be obtained due to plotting limitations. The algebraic methods, such as substitution and elimination, provide precise numerical solutions without graphing. These methods are generally more accurate and efficient for examination purposes. While the graphical method helps understand concepts visually, algebraic methods are preferred when exact answers are required. Both methods are important and complement each other in solving linear equations.
Q17. What is a consistent and independent system?
Answer:
A consistent and independent system is a pair of linear equations that has exactly one solution. This occurs when the two lines intersect at a single point. Algebraically, the condition is:
[\frac{a_1}{a_2}\ne\frac{b_1}{b_2}]
Since a common solution exists, the system is called consistent. Since neither equation is a multiple of the other, the equations are independent. Such systems are the most common type encountered in mathematics. The unique intersection point represents the only ordered pair satisfying both equations simultaneously. These systems can be solved using graphical, substitution, or elimination methods.
Q18. What is a consistent and dependent system?
Answer:
A consistent and dependent system has infinitely many solutions. This happens when both equations represent the same straight line. The algebraic condition is:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}]
Since at least one solution exists, the system is consistent. Since one equation depends on the other and can be obtained by multiplying it by a constant, the equations are dependent. Every point on the common line satisfies both equations. Therefore, there are infinitely many solutions. Such systems are represented graphically by coincident lines.
Q19. What is an inconsistent system?
Answer:
An inconsistent system is a pair of linear equations that has no solution. The equations represent parallel lines which never intersect. The algebraic condition for inconsistency is:
[\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}]
Since there is no common point between the lines, no values of (x) and (y) satisfy both equations simultaneously. Such systems often arise when two equations have the same slope but different intercepts. Recognizing an inconsistent system helps students determine the nature of solutions without performing lengthy calculations.
Q20. Why is Chapter 3 important for board examinations?
Answer:
Chapter 3 is an important chapter in CBSE Class 10 Mathematics because it introduces methods for solving simultaneous equations and develops logical reasoning skills. Students learn graphical representation, consistency conditions, substitution method, elimination method, and application-based problems. Questions from this chapter frequently appear in board examinations as short-answer and application-based questions. The concepts also form the foundation for higher studies in algebra, coordinate geometry, and calculus. Mastering this chapter helps students solve real-life mathematical situations effectively and improves problem-solving ability. Regular practice of numerical and word problems is essential for scoring well in examinations.