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NCERT Solutions · Class 10 · Chapter 3: Pair of Linear Equations in Two Variables

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.3

Exercise 3.3: Solving by elimination. Multiply the equations so one variable has equal (or opposite) coefficients, add or subtract to eliminate it, solve, then substitute back.

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Exercise 3.3 questions and solutions

Exercise 3.3, Question 1

Solve by elimination (the substitution method gives the same answers).
(i) \(x + y = 5,\ 2x - 3y = 4\)
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  1. \(3 \times\) first \(+\) second: \(5x = 19 \Rightarrow x = \tfrac{19}{5}\).
  2. \(y = 5 - \tfrac{19}{5} = \tfrac65\).
Answer: \(x = \tfrac{19}{5},\ y = \tfrac65\)
(ii) \(3x + 4y = 10,\ 2x - 2y = 2\)
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  1. First \(+ 2 \times\) second: \(7x = 14 \Rightarrow x = 2\).
  2. \(y = x - 1 = 1\).
Answer: \(x = 2,\ y = 1\)
(iii) \(3x - 5y - 4 = 0,\ 9x = 2y + 7\)
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  1. \(3 \times\) first: \(9x - 15y = 12\); subtract \(9x - 2y = 7\): \(-13y = 5\).
  2. \(y = -\tfrac{5}{13}\), \(x = \tfrac{4 + 5y}{3} = \tfrac{9}{13}\).
Answer: \(x = \tfrac{9}{13},\ y = -\tfrac{5}{13}\)
(iv) \(\tfrac x2 + \tfrac{2y}{3} = -1,\ x - \tfrac y3 = 3\)
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  1. Clear fractions: \(3x + 4y = -6\), \(3x - y = 9\).
  2. Subtract the second from the first: \(5y = -15 \Rightarrow y = -3\).
  3. Then \(3x = 9 + y = 6 \Rightarrow x = 2\). Check: \(3(2) + 4(-3) = -6\) ✓
Answer: \(x = 2,\ y = -3\)

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Exercise 3.3, Question 2

Form the pair of equations and solve by elimination.
(i) Adding 1 to the numerator and subtracting 1 from the denominator makes a fraction 1; adding 1 only to the denominator makes it \(\tfrac12\).
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  1. \(x + 1 = y - 1\) and \(2x = y + 1\): \(x - y = -2\), \(2x - y = 1\).
  2. Subtract: \(x = 3\), so \(y = 5\).
Answer: \(\tfrac35\)
(ii) Five years ago A was three times as old as B; in ten years A will be twice as old as B.
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  1. \(x - 5 = 3(y - 5)\) and \(x + 10 = 2(y + 10)\): \(x - 3y = -10\), \(x - 2y = 10\).
  2. Subtract: \(y = 20\), \(x = 50\).
Answer: A is 50, B is 20
(iii) The digits of a two-digit number add to 9, and nine times the number is twice the number with its digits reversed.
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  1. Number \(10x + y\): \(x + y = 9\) and \(9(10x + y) = 2(10y + x)\), i.e. \(88x - 11y = 0 \Rightarrow y = 8x\).
  2. \(9x = 9 \Rightarrow x = 1,\ y = 8\).
Answer: \(18\)
(iv) ₹2000 is withdrawn in 25 notes, all ₹50 or ₹100. How many of each?
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  1. \(x + y = 25\), \[\begin{aligned}&50x + 100y = 2000 \\ \Rightarrow\ &x + 2y = 40\end{aligned}\]
  2. Subtract: \(y = 15\), \(x = 10\).
Answer: 10 notes of ₹50 and 15 of ₹100
(v) A library charges a fixed amount for the first three days and a daily rate after that. 7 days cost ₹27 and 5 days ₹21. Find both charges.
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  1. \(x + 4y = 27\), \(x + 2y = 21\).
  2. Subtract: \(2y = 6 \Rightarrow y = 3\), \(x = 15\).
Answer: Fixed ₹15, then ₹3 a day

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Pair of Linear Equations in Two Variables in our sample papers: Sample paper 1 (questions 3, 27) · Sample paper 2 (questions 3, 22) · Sample paper 3 (questions 4, 27) · Sample paper 4 (questions 4, 27) · Sample paper 5 (questions 3, 32, 38).

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Textbook: NCERT Mathematics Class 10 (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.