If a pair of linear equations in two variables is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, and a1/a2 ≠ b1/b2, then which of the following is true about the pair?

It has infinitely many solutions.

To solve the equations x + y = 7 and y = 2x + 1 using substitution, which equation do we get after substituting y in the first equation?

Both 3x + 1 = 7 and x + 2x + 1 = 7

Assertion (A): The pair of equations 2x + 5y = 10 and 4x + 10y = 20 are inconsistent. Reason (R): If a1/a2 = b1/b2 = c1/c2, the system is consistent.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

Solve the following pair of linear equations graphically: 3x + 4y = 10 and 2x - 2y = 2. Also, write the coordinates of the point where the lines intersect the x-axis and y-axis.

The solution is x = 2, y = 1. The line 3x+4y=10 meets x-axis at (10/3,0) and y-axis at (0,2.5); the line 2x-2y=2 meets x-axis at (1,0) and y-axis at (0,-1).

A motor boat covers 25 km upstream and 39 km downstream in 8 hours. In 11 hours, it can cover 35 km upstream and 52 km downstream. Find the speed of the boat in still water and the speed of the stream.

Speed of boat in still water = 9 km/h, speed of stream = 4 km/h.

The sum of the present ages of Anil and Sunil is 25 years. Five years ago, Anil's age was twice Sunil's age at that time. (1) Form the pair of linear equations representing the given situation. (2) Determine their present ages.

Anil is 15 years old and Sunil is 10 years old.

Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables — Important Questions & Sample Paper

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Class 10 Mathematics Chapter 3, "Pair of Linear Equations in Two Variables," introduces students to simultaneous linear equations and their solutions. The chapter covers two-variable linear equations and how to represent them graphically as straight lines. Students learn three algebraic methods: substitution, elimination, and cross-multiplication, each applicable depending on the form of the equations. The graphical method shows whether lines intersect (unique solution), are parallel (no solution), or coincide (infinite solutions). Conditions for consistency are derived using ratios of coefficients (a1/a2, b1/b2, c1/c2). The chapter also includes a wide range of word problems: fractions, ages, two-digit numbers, time and work, boat and stream, and geometry-based applications. Typical exam questions involve solving a system by elimination or substitution, finding the value of k for a specific type of solution, reducing equations to linear form (e.g., 1/x, 1/y), and setting up equations from real-life contexts like upstream/downstream speed. Mastering this chapter builds algebraic fluency and problem-solving skills essential for CBSE board exams.

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Mathematics — Pair of Linear Equations in Two Variables

Class 10Time: 3 hrsMax Marks: 80

SECTION A

1.
If a pair of linear equations in two variables is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, and a1/a2 ≠ b1/b2, then which of the following is true about the pair?
(a) It has no solution.(b) It has a unique solution.(c) It has infinitely many solutions.(d) It has two solutions.
1
2.
To solve the equations x + y = 7 and y = 2x + 1 using substitution, which equation do we get after substituting y in the first equation?
(a) 3x + 1 = 7(b) 3x + 2 = 7(c) x + 2x + 1 = 7(d) Both 3x + 1 = 7 and x + 2x + 1 = 7
1
3.
A fraction becomes 1/2 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is added to its denominator. What is the fraction?
(a) 3/5(b) 4/7(c) 5/9(d) No such fraction exists
1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

Real, exam-style questions to practise and revise — each with its answer. Generate a full paper for unlimited more.

Q1. If a pair of linear equations in two variables is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, and a1/a2 ≠ b1/b2, then which of the following is true about the pair?
1 mark
Multiple Choice (MCQ)
(A) It has no solution.(B) It has a unique solution.(C) It has infinitely many solutions.(D) It has two solutions.
▸ Answer▾ Answer
It has a unique solution.
Q2. To solve the equations x + y = 7 and y = 2x + 1 using substitution, which equation do we get after substituting y in the first equation?
1 mark
Multiple Choice (MCQ)
(A) 3x + 1 = 7(B) 3x + 2 = 7(C) x + 2x + 1 = 7(D) Both 3x + 1 = 7 and x + 2x + 1 = 7
▸ Answer▾ Answer
Both 3x + 1 = 7 and x + 2x + 1 = 7
Q3. A fraction becomes 1/2 when 2 is subtracted from the numerator and it becomes 1/2 when 1 is added to its denominator. What is the fraction?
1 mark
Multiple Choice (MCQ)
(A) 3/5(B) 4/7(C) 5/9(D) No such fraction exists
▸ Answer▾ Answer
No such fraction exists
Q4. If (x, y) is the solution of the system 4x + 3y = 17 and 2x − y = 1, then x + y equals
1 mark
Multiple Choice (MCQ)
(A) 4(B) 5(C) 6(D) 7
▸ Answer▾ Answer
5
Q5. Assertion (A): The pair of equations 2x + 5y = 10 and 4x + 10y = 20 are inconsistent. Reason (R): If a1/a2 = b1/b2 = c1/c2, the system is consistent.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
A is false but R is true.
Q6. Solve the following pair of linear equations by substitution method: x + y = 5, 2x – 3y = 4.
2 marks
Short Answer
▸ Answer▾ Answer
x = 19/5, y = 6/5
Q7. The sum of the digits of a two-digit number is 9. Nine times this number is twice the number obtained by reversing its digits. Find the number.
2 marks
Short Answer
▸ Answer▾ Answer
18
Q8. Find the value of k for which the following system of equations has no solution: kx + 3y = k - 3, 12x + ky = k.
3 marks
Short Answer
▸ Answer▾ Answer
k = -6.
Q9. Solve the following pair of linear equations using the elimination method: 41x + 53y = 135, 53x + 41y = 147.
3 marks
Short Answer
▸ Answer▾ Answer
x = 2, y = 1.
Q10. Solve the following pair of linear equations graphically: 3x + 4y = 10 and 2x - 2y = 2. Also, write the coordinates of the point where the lines intersect the x-axis and y-axis.
5 marks
Long Answer
▸ Answer▾ Answer
The solution is x = 2, y = 1. The line 3x+4y=10 meets x-axis at (10/3,0) and y-axis at (0,2.5); the line 2x-2y=2 meets x-axis at (1,0) and y-axis at (0,-1).
Q11. A motor boat covers 25 km upstream and 39 km downstream in 8 hours. In 11 hours, it can cover 35 km upstream and 52 km downstream. Find the speed of the boat in still water and the speed of the stream.
5 marks
Long Answer
▸ Answer▾ Answer
Speed of boat in still water = 9 km/h, speed of stream = 4 km/h.
Q12. The sum of the present ages of Anil and Sunil is 25 years. Five years ago, Anil's age was twice Sunil's age at that time.
4 marks
Case Study
1. (i) Form the pair of linear equations representing the given situation.2 marks
2. (ii) Determine their present ages.2 marks
▸ Answer▾ Answer
Anil is 15 years old and Sunil is 10 years old.

Frequently asked questions

How do I know which method—substitution, elimination, or cross-multiplication—to use for solving a pair of linear equations?

Choose based on the coefficients. Elimination is often quickest when coefficients are easy to align, e.g., if one variable has opposite coefficients. Substitution works well when one equation is already solved for one variable or has a coefficient of 1. Cross-multiplication is a direct formula method useful when both equations are in standard form ax+by+c=0, especially with complex coefficients.

What are the conditions for a system to have a unique solution, infinite solutions, or no solution?

Using the ratios of coefficients: if a1/a2 ≠ b1/b2, the lines intersect (unique solution). If a1/a2 = b1/b2 = c1/c2, the lines coincide (infinitely many solutions). If a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel (no solution). CBSE often asks to find k such that a system satisfies one of these conditions.

How do I set up equations for upstream and downstream problems?

Assume speed of boat in still water = u km/h, speed of stream = v km/h. Downstream speed = u+v, upstream speed = u−v. Time = distance/speed. Use the given times and distances to form two equations: e.g., distance1/(u+v) + distance2/(u−v) = total time. Solve these linear equations by putting 1/(u+v) and 1/(u−v) as variables.

Can I get practice questions similar to real CBSE exams from this chapter?

Yes, on QPaper.in you can generate custom question papers with problems exactly like those in CBSE exams: solving by elimination, finding k for infinite/no solution, word problems on fractions, ages, and boats. The question bank includes a variety of difficulty levels and marks distribution.

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