This post is a complete practice guide for System of Two Linear Equations in Two Variables, one of the most important chapters in CBSE Class 10 Mathematics. It is carefully designed to help students score better in board exams by covering unique solution, no solution, and infinitely many solutions with a wide range of exam-oriented questions. The content is equally useful for ICSE Mathematics, State Board exams, KV and Navodaya syllabus, and other school-level competitive exams. This post includes word problems, tricky questions, and parameter-based problems, which are highly useful for building conceptual clarity and improving problem-solving skills. It also serves as a strong foundation for NTSE preparation, Olympiad maths, and competitive exam coaching. With step-by-step practice and CBSE pattern questions, this resource is ideal for online maths learning, home tuition support, and self-study. Overall, this post works as an all-in-one maths practice material for students aiming for high marks and long-term success in mathematics.
Solved Example 1 (Unique Solution)
Solve the system: \[ x + y = 7,\quad x - y = 1 \]
Adding both equations:
\[ (x+y)+(x-y)=7+1 \] \[\Rightarrow 2x=8 \Rightarrow x=4 \]
Substitute \(x=4\) in \(x+y=7\):
\[ 4+y=7 \Rightarrow y=3 \]
Answer: \(x=4,\; y=3\) (Unique solution)
Solved Example 2 (Unique Solution with Fractions)
Solve: \[ \frac{x}{2}+\frac{y}{3}=4,\quad \frac{x}{2}-\frac{y}{3}=2 \]
Multiply both equations by 6 to remove fractions:
\[ 3x+2y=24,\quad 3x-2y=12 \]
Adding: \[ 6x=36 \Rightarrow x=6 \]
Substitute \(x=6\) in \(3x+2y=24\):
\[ 18+2y=24 \Rightarrow y=3 \]
Answer: \(x=6,\; y=3\)
Solved Example 3 (No Solution – Parallel Lines)
Determine the nature of the solution: \[ 2x+3y=7,\quad 4x+6y=15 \]
Compare coefficients:
\[ \frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\quad \frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2},\quad\] \[ \frac{c_1}{c_2}=\frac{7}{15}\neq\frac{1}{2} \]
Since \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\), the lines are parallel.
Answer: No solution
Solved Example 4 (Infinitely Many Solutions)
Find the nature of the system: \[ 3x+2y=10,\quad 6x+4y=20 \]
Compare coefficients:
\[ \frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2},\quad \frac{b_1}{b_2}=\frac{2}{4}=\frac{1}{2},\quad\] \[ \frac{c_1}{c_2}=\frac{10}{20}=\frac{1}{2} \]
All ratios are equal, so both equations represent the same line.
Answer: Infinitely many solutions
Solved Example 5 (Word Problem)
The sum of two numbers is 20 and their difference is 4. Find the numbers.
Let the numbers be \(x\) and \(y\).
\[ x+y=20 \quad (1) \]
\[ x-y=4 \quad (2) \]
Adding (1) and (2):
\[ 2x=24 \Rightarrow x=12 \]
Substitute in \(x+y=20\):
\[ 12+y=20 \Rightarrow y=8 \]
Answer: The numbers are \(12\) and \(8\)
Conditions for Pair of Linear Equations in Two Variables
Consider the pair of linear equations:
\[
a_1x + b_1y + c_1 = 0
\]
\[
a_2x + b_2y + c_2 = 0
\]
1️⃣ Unique Solution (Intersecting Lines)
The given pair of linear equations has a unique solution if:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
In this case, the two lines intersect at exactly one point.
2️⃣ No Solution (Parallel Lines)
The given pair of linear equations has no solution if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
In this case, the two lines are parallel and never intersect.
3️⃣ Infinitely Many Solutions (Coincident Lines)
The given pair of linear equations has infinitely many solutions if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
In this case, both equations represent the same line.
These conditions are extremely important for CBSE Class 10 Mathematics and are frequently tested in board examinations, case-study based questions, and competitive exams.
Q.1
Find \(x\) and \(y\) if
\(x + y = 9,\; x - y = 3\)
Q.2
Find \(x\) and \(y\) if \(2x + y = 11,\; x + y = 8\)
Q.3
Find \(x\) and \(y\) if
\(3x - y = 7,\; x + y = 5\)
Q.4
Find \(x\) and \(y\) if
\(2x - y = 4,\; x + y = 7\)
Q.5
\(4x + y = 13,\; 2x - y = 1\)
Q.6
\(3x + 2y = 12,\; x + 2y = 8\)
Q.7
\(5x - y = 14,\; 3x + y = 10\)
Q.8
\(2x + 3y = 13,\; x + y = 6\)
Q.9
\(4x - 3y = 5,\; 2x + y = 7\)
Q.10
\(x + ky = 4,\; 2x + 6y = 8\)
Q.11
For what value of \(k\) \(kx + y = 3,\; 2x + 2y = 6\) has unique solution
Q.12
\(3x - 2y = 6,\; 6x - 4y = 15\)
Q.13
\(2x + 3y = 12,\; 4x + 6y = 18\)
Q.14
Sum of two numbers is 12 and Difference is, find the numbers 4
Q.15
Sum of two numbers is 25 and Difference is 5, find the numbers
Q.16
Cost problem (notebooks & pens)
Q.17
Perimeter of a rectangle is 48 cm, while Length is 4 more then Breadth, find both
Q.18
Sum of two digits = 11 and Difference = 3, find numbers.
Q19.
The present age of a father is three times the present age of his son. After 10 years, the sum of their ages will be 70 years. Find the present ages of the father and the son.
Q.20
Sum of Twice a number and other number is 20 and Difference is 4, find numbers.
Q.21
Solve the system: \( \frac{x}{2}+\frac{y}{3}=5,\quad x-\frac{y}{2}=1 \)
Q.22
Solve: \( \frac{x}{3}-\frac{y}{4}=1,\quad \frac{x}{2}+\frac{y}{4}=4 \)
Q.23
Solve: \( \frac{2x+y}{3}=4,\quad \frac{x-y}{2}=1 \)
Q.24
Solve: \[ \frac{x+y}{2}=4,\quad \frac{x-y}{3}=1 \]
Q.25
Solve: \( \frac{3x-2y}{4}=1,\quad \frac{x+y}{2}=3 \)
Q.26
Solve: \( \frac{x}{4}+\frac{y}{6}=2,\quad \frac{x}{2}-\frac{y}{3}=1 \)
Q.27
Solve: \( \frac{x-1}{2}+\frac{y+3}{3}=4,\quad x+y=5 \)
Q.28
Solve: \( 0.5x+0.25y=3,\quad x-y=2 \)
Q.29
Solve: \( \frac{x}{5}+\frac{y}{2}=3,\quad \frac{x}{2}-\frac{y}{5}=1 \)
Q.30
Solve: \( \frac{x}{4}-\frac{y}{6}=1,\quad \frac{x}{2}+\frac{y}{3}=5 \)
Q.30
The sum of two numbers is 20 and their difference is 4. Find the numbers.
Q.31
Two numbers are such that twice the first added to the second gives 26, and the first exceeds the second by 2. Find the numbers.
Q.32
The sum of the digits of a two-digit number is 10. The number formed by reversing the digits is 36 more than the original number. Find the number.
Q.33
The cost of 3 notebooks and 2 pens is ₹86. The cost of 5 notebooks and 4 pens is ₹154. Find the cost of one notebook and one pen.
Q.34
The perimeter of a rectangle is 52 cm. If the length is 4 cm more than the breadth, find its dimensions.
Q.35
The sum of the ages of two persons is 36 years. Five years ago, one person was twice as old as the other. Find their present ages.
Q.36
A boat travels 24 km downstream in 2 hours and upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream.
Q.37
Two angles are supplementary. One angle is 18° more than the other. Find the angles.
Q.38
The sum of two numbers is 25. One number is 5 more than the other. Find the numbers.
Q.39
The difference between two numbers is 6 and their sum is 30. Find the numbers.
Q.40
The cost of 2 chairs and 3 tables is ₹2900. The cost of 4 chairs and 1 table is ₹2600. Find the cost of one chair and one table.
Q.41
The sum of two consecutive integers is 47. Find the integers.
Q.42
The sum of two numbers is 18 and one number is twice the other. Find the numbers.
Q.43
The length of a rectangle is 6 cm more than its breadth. If the perimeter is 52 cm, find its dimensions.
Q.44
A fraction becomes \(\frac{2}{3}\) when 1 is added to both numerator and denominator. Find the fraction.
Q.45
The sum of two numbers is 50 and their difference is 10. Find the numbers.
Q.46
A cinema sold 120 tickets for a show. Adult tickets cost ₹50 and child tickets cost ₹30. The total collection was ₹5200. Find the number of adult and child tickets sold.
Q.47
Two numbers are such that three times the first plus twice the second is 29, and the sum of the numbers is 11. Find the numbers.
Q.48
In a class, the number of boys is 6 more than the number of girls. The total number of students is 42. Find the number of boys and girls.
Q.49
A shopkeeper buys apples and oranges for ₹500. The cost of an apple is ₹10 and that of an orange is ₹5. If the total number of fruits bought is 70, find how many apples and oranges were bought.
Q.50
The sum of two numbers is 16. If the first number is decreased by 2 and the second is increased by 4, their sum becomes 18. Find the numbers.
Q.51
A two-digit number is such that the sum of its digits is 9. If the digits are interchanged, the new number is 27 more than the original number. Find the number.
Q.52
The perimeter of a rectangle is 60 cm. If the length is twice the breadth minus 4 cm, find the dimensions of the rectangle.
Q.53
The sum of the ages of a father and his son is 50 years. Ten years ago, the father was four times as old as his son. Find their present ages.
Q.54
Two numbers are such that one-fourth of the first plus one-third of the second is 7, and the sum of the numbers is 36. Find the numbers.
Q.55
A train covers 300 km at a certain speed. If the speed were increased by 5 km/h, it would take 2 hours less. Find the original speed of the train.
Q.56
A boat travels 24 km downstream in 2 hours and the same distance upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream.
Q.57
A boat goes 30 km downstream in 3 hours and returns upstream in 5 hours. Find the speed of the boat in still water and the speed of the stream.
Q.58
A boat covers 20 km downstream in 2 hours and the same distance upstream in 5 hours. Find the speed of the boat in still water and the speed of the stream.
Q.59
A boat takes 1 hour less to travel 18 km downstream than to travel the same distance upstream. If the speed of the stream is 2 km/h, find the speed of the boat in still water.
Q.60
A boat travels 40 km downstream in 4 hours and upstream in 8 hours. Find the speed of the boat in still water and the speed of the stream.
Q.61
In a school canteen, the price of 2 burgers and 3 sandwiches is ₹190. The price of 3 burgers and 1 sandwich is ₹170. Find the price of one burger and one sandwich.
Q.62
A mobile recharge shop sold two types of plans: ₹199 and ₹299. A total of 40 plans were sold for ₹10,360. How many plans of each type were sold?
Q.63
A shopkeeper sells apples at ₹20 each and oranges at ₹10 each. If 50 fruits were sold for a total of ₹800, find the number of apples and oranges sold.
Q.64
Two numbers are such that five times the first minus twice the second is 4, and three times the first plus the second is 29. Find the numbers.
Q.65
The sum of two numbers is 14. If one number is multiplied by 3 and the other by 2, their sum becomes 36. Find the numbers.
Q.66
A taxi charges a fixed fare plus a per-kilometre charge. A ride of 10 km costs ₹160, while a ride of 18 km costs ₹256. Find the fixed fare and the charge per km.
Q.67
A two-digit number is such that the sum of its digits is 11. If the digits are reversed, the new number is 27 less than the original. Find the number.
Q.68
A rectangle has perimeter 64 cm. If the length is 4 cm less than three times the breadth, find its dimensions.
Q.69
The sum of the ages of a mother and her daughter is 42 years. Five years ago, the mother was three times as old as the daughter. Find their present ages.
Q.70
A fraction becomes \(\frac{3}{5}\) when 2 is added to both its numerator and denominator. Find the fraction.
Q.71
Determine the nature of the solution: \[ 2x+3y=7,\quad 4x+6y=14 \]
Q.72
Find the nature of the system: \[ 3x-2y=5,\quad 6x-4y=9 \]
Q.73
Determine the number of solutions: \[ x+y=6,\quad x-y=2 \]
Q.74
Check consistency: \[ 4x+5y=10,\quad 8x+10y=20 \]
Q.75
Determine the nature: \[ 5x-3y=7,\quad 10x-6y=15 \]
Q.76
Find the nature of solution: \[ 2x+y=4,\quad x-y=1 \]
Q.77
Check whether the system has a solution: \[ 7x+3y=21,\quad 14x+6y=42 \]
Q.78
Determine the nature: \[ x+2y=5,\quad 2x+4y=11 \]
Q.79
Check consistency: \[ 3x+y=8,\quad x+3y=8 \]
Q.80
Find the nature: \[ 4x-y=9,\quad 8x-2y=18 \]
Q.81
Determine the nature: \[ 2x-3y=4,\quad 4x-6y=10 \]
Q.82
Check whether the system has a unique solution: \[ x+3y=11,\quad 2x+5y=18 \]
Q.83
Find the nature: \[ 6x+2y=14,\quad 3x+y=7 \]
Q.84
Determine consistency: \[ 5x+y=9,\quad 10x+2y=20 \]
Q.85
Check nature of solution: \[ 3x-4y=5,\quad x+y=6 \]
Q.86
Find the nature: \[ 2x+5y=10,\quad 4x+10y=20 \]
Q.87
Determine the nature: \[ 7x-2y=9,\quad 14x-4y=20 \]
Q.88
Check consistency: \[ x-y=1,\quad 2x+y=8 \]
Q.89
Determine the nature: \[ 9x+3y=12,\quad 3x+y=4 \]
Q.90
Check whether the system has a solution: \[ 4x+7y=11,\quad 8x+14y=25 \]
Q.91
Find the value of \(k\) for which the system has infinitely many solutions:
\(\quad\quad
(k-1)x + 2y = 3\)
\(\quad\quad 2x + (4-2k)y = 6
\)
Q.92
For what value of \(k\) does the system have unique solution ? \[ 3x + ky = 5,\quad 6x + 4y = 10 \]
Q.93
Determine the nature of the solution: \[ \frac{2x-3y}{5} = 1,\quad \frac{4x-6y}{10} = 2 \]
Q.94
Find the value of \(k\) for which the system has a unique solution: \[ kx + y = 3,\;x + ky = 3 \]
Q.95
Determine the nature of the system:
\(\quad\quad
(2k+1)x + 3y = 6\)
\(\quad\quad 4x + (6k-1)y = 12
\)
Q.96
For what value of \(k\) does the following system represent parallel lines? \[ kx + 2y = 4,\; 6x + (k+1)y = 12 \]
Q.97
Find the value of \(k\) for which the system has infinitely many solutions:
\(
(k+2)x + 4y = 8\),
\(2x + (k-2)y = 4
\)
Q.98
Determine the nature of the solution: \[ \frac{x}{2} - \frac{y}{3} = 1,\; \frac{3x - 2y}{6} = 2 \]
Q.99
For what value of \(k\) does the system have no solution?
\(\quad\quad
(k-3)x + 2y = 5,\)
\(\quad 2x + (k-1)y = 10
\)
Q.100
Find the value of \(k\) for which the system has infinitely many solutions:
\(\quad\quad
(k-1)x + 2y = 4,\)
\(\quad\quad 2x + (k-1)y = 4
\)
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