Pair of Linear Equations in Two Variables Class 10 Maths Worksheet with Answers Chapter 3

Pair of Linear Equations in Two Variables is the 3rd chapter in NCERT class 10 mathematics book. The worksheet below is split into four levels: Basic, Standard, Advance, and HOTS. Check the Answer Key hints only after you’ve tried each question on your own. Below are some of the major sections covered in the chapter.

• Linear equations basics
• Graphical solution method
• Substitution and elimination
• Consistency conditions rules
• Word problems modelling

Class 10 Maths Worksheet – Chapter 3: Pair of Linear Equations in Two Variables

Basic

1. Write the general form of a linear equation in two variables.
   Format: a x + b y + c = 0
2. Solve by graphical idea (write the solution point only):
   x + 3y = 6
   2x − 3y = 12
3. For the pair of lines, write whether they are intersecting, parallel or coincident:
   2x + 3y − 9 = 0
   4x + 6y − 18 = 0
4. Check whether the pair is consistent or inconsistent:
   x + 2y − 4 = 0
   2x + 4y − 12 = 0
5. Find two solutions for each equation:
   x + y = 5
   x − y = 1
6. True/False (write one-line reason):
   • If two lines intersect, the pair has exactly one solution.
   • If two lines are parallel, the pair has infinitely many solutions.
   • If two lines coincide, the pair has infinitely many solutions.
7. Form a pair of linear equations for:
   “The number of rides is x and the number of hoopla games is y.
   Hoopla games are half the rides, and total cost is ₹20 if each ride costs ₹3 and each game costs ₹4.”
   Write the two equations.

Standard

1. Solve by substitution method:
   x + y = 14
   x − y = 4
2. Solve by substitution method:
   0.2x + 0.3y = 1.3
   0.4x + 0.5y = 2.3
3. Solve by elimination method:
   3x + 2y = 5
   2x − 3y = 7
4. Solve by elimination method:
   2x + 3y = 11
   2x − 4y = −24
5. Find the value of m such that the line y = mx + 3 passes through the solution of:
   2x + 3y = 11
   2x − 4y = −24
6. Compare ratios and decide whether the pair has:
   no solution / unique solution / infinitely many solutions
   5x − 8y + 1 = 0
   3x − (24/5)y + (3/5) = 0
7. The sum of a two-digit number and the number formed by reversing the digits is 66.
   The digits differ by 2.
   Find the numbers.

Advance

1. Solve by substitution:
   7x − 15y = 2
   x + 2y = 3
   (Give the solution in fraction form if needed.)
2. Solve by elimination:
   3x + 4y = 10
   2x − 2y = 2
3. Check consistency using ratios:
   2x − 3y = 8
   4x − 6y = 9
   Then state: consistent or inconsistent.
4. A fraction becomes 9/11 if 2 is added to both numerator and denominator.
   It becomes 5/6 if 3 is added to both numerator and denominator.
   Find the fraction.
5. Taxi charges: For 10 km, fare is ₹105. For 15 km, fare is ₹155.
   Find:
   • Fixed charge
   • Charge per km
   • Fare for 25 km
6. Half the perimeter of a rectangle is 36 m.
   Length is 4 m more than width.
   Find length and width.
7. Draw the graphs of:
   x − y + 1 = 0
   3x + 2y − 12 = 0
   Find the vertices of the triangle formed by these lines and the x-axis.

HOTS

1. Without solving fully, predict the nature of solutions using ratios:
   4x + 6y − 8 = 0
   2x + 3y − 4 = 0
   Write: unique / none / infinite, with reason.
2. A pair of equations has infinitely many solutions.
   Write one example of such a pair and explain in one line why.
3. A pair of equations has no solution.
   Write one example of such a pair and explain in one line why.
4. The lines 2x + 3y = 12 and 4x + 6y = k are:
   • parallel for which values of k?
   • coincident for which values of k?
5. If the solution of a pair is (3, −2), form one pair of linear equations having this solution.
   (Give any one correct pair.)
6. Solve and interpret:
   9x − 4y = 2000
   7x − 3y = 2000
   Then write the monthly incomes and expenditures as per ratio interpretation.
7. A student solved:
   2x + 3y = 9
   4x + 6y = 18
   and wrote “no solution”.
   Is the student correct? Explain the mistake in 2–3 lines.

Answer Key

Basic – Answers

1. Ans: a x + b y + c = 0, where a and b are not both 0.
   Hint: At least one of a or b must be non-zero.
2. Ans: (6, 0)
   Hint: The intersection point is the solution. (Shown in the chapter example.)
3. Ans: Coincident
   Hint: Second equation is exactly 2 times the first.
4. Ans: Inconsistent (no solution)
   Hint: Coefficients of x and y are in same ratio but constants are not.
5. Ans:
   • For x + y = 5: (0,5), (5,0)
   • For x − y = 1: (1,0), (2,1)

   Hint: Put x = 0 and y = 0 to get two points.

6. Ans:
   • True — intersecting lines meet at one point.
   • False — parallel lines never meet, so no solution.
   • True — coincident lines overlap, so infinite solutions.

   Hint: Link “type of lines” to “number of solutions”.

7. Ans: y = x/2 and 3x + 4y = 20
   Hint: “Half the rides” means y = x/2.

Standard – Answers

1. Ans: x = 9, y = 5
   Hint: From x − y = 4 ⇒ x = y + 4, substitute in x + y = 14.
2. Ans: Multiply by 10:
   2x + 3y = 13
   4x + 5y = 23
   Solution: x = 2, y = 3
   Hint: Clear decimals first.
3. Ans: x = 29/13, y = −11/13
   Hint: Use elimination: make y coefficients equal and subtract.
4. Ans: x = −3, y = 5
   Hint: Subtract equations to eliminate x.
5. Ans: From previous solution x = −3, y = 5
   5 = m(−3) + 3 ⇒ m = −2/3
   Hint: Substitute the solution point into y = mx + 3.
6. Ans: Infinitely many solutions
   Hint: One equation becomes the other after multiplying by a constant.
7. Ans: The numbers are 42 and 24.
   Hint: Let digits be x and y, use (10x + y) + (10y + x) = 66 and |x − y| = 2.

Advance – Answers

1. Ans: x = 49/29, y = 19/29
   Hint: From x + 2y = 3, write x = 3 − 2y and substitute.
2. Ans: x = 2, y = 1
   Hint: Multiply second equation by 2 to eliminate y.
3. Ans: Inconsistent (no solution)
   Hint: a₁/a₂ = b₁/b₂ but c₁/c₂ is different.
4. Ans: Let fraction = x/y
   (x + 2)/(y + 2) = 9/11 ⇒ 11x − 9y = −4
   (x + 3)/(y + 3) = 5/6 ⇒ 6x − 5y = −3
   Solving gives x = 1, y = 4 ⇒ fraction = 1/4
   Hint: Cross-multiply both equations, then solve.
5. Ans: Fixed + 10r = 105, Fixed + 15r = 155
   Subtract: 5r = 50 ⇒ r = 10
   Fixed = 105 − 100 = 5
   For 25 km: 5 + 25×10 = ₹255
   Hint: Let per-km charge be r and fixed be F.
6. Ans: Let width = w, length = w + 4
   Half perimeter: (l + w) = 36 ⇒ (w + 4) + w = 36 ⇒ 2w = 32 ⇒ w = 16
   Length = 20
   Hint: Half perimeter equals l + w.
7. Ans: For x − y + 1 = 0 ⇒ y = x + 1; x-intercept at (−1, 0)
   For 3x + 2y − 12 = 0 ⇒ y = (12 − 3x)/2; x-intercept at (4, 0)
   y-axis intercepts: (0, 1) and (0, 6)
   Triangle vertices: (−1, 0), (4, 0), (0, 1) or (0, 6) depending on region
   Hint: Use intercepts with axes to get vertices.

HOTS – Answers

1. Ans: Infinite solutions
   Hint: Second equation is exactly 2 times the first ⇒ coincident.
2. Ans: Example: x + y = 5 and 2x + 2y = 10
   Hint: One equation is a multiple of the other.
3. Ans: Example: x + y = 5 and x + y = 7
   Hint: Same left side, different constants ⇒ parallel lines.
4. Ans:
   • Parallel when k ≠ 24
   • Coincident when k = 24

   Hint: 4x + 6y = 24 matches 2×(2x + 3y = 12).

5. Ans: One pair: x + y = 1 and x − y = 5
   Hint: Choose any two lines intersecting at (3, −2).
6. Ans: Multiply first by 3 and second by 4:
   27x − 12y = 6000
   28x − 12y = 8000
   Subtract: x = 2000
   Then 9x − 4y = 2000 ⇒ 18000 − 4y = 2000 ⇒ y = 4000
   Incomes: ₹9x = ₹18000 and ₹7x = ₹14000
   Hint: Make coefficients equal, then subtract.
7. Ans: Not correct.
   The equations are multiples of each other, so lines coincide.
   Hence there are infinitely many solutions, not “no solution”.
   Hint: If you get a true statement like 18 = 18, it means infinite solutions.

Worksheet for Other chapters

• Real Numbers Class 10 Maths Worksheet
• Polynomials Class 10 Maths Worksheet
• Quadratic Equations Class 10 Maths Worksheet
• Arithmetic Progressions Class 10 Maths Worksheet
• Introduction to Trigonometry Class 10 Maths Worksheet
• Some Applications of Trigonometry Class 10 Maths Worksheet
• Areas Related to Circles Class 10 Maths Worksheet
• Probability Class 10 Maths Worksheet