Exercise 3.3 answers: Pair of Linear Equations in Two Variables

Class 10 Maths2 questions

Exercise 3.3

2 questions · page 36 of the book

Question 1

“Solve the following pair of linear equations by the elimination method and the substitution method” · p. 36

Open NCERT p. 36Matches NCERT’s answer

(i) x + y = 5 and 2x − 3y = 4

  1. Elimination: multiply x + y = 5 by 3 to get 3x + 3y = 15. Add it to 2x − 3y = 4: 5x = 19, so x = 19/5. Then y = 5 − 19/5 = 6/5.
  2. Substitution: from x + y = 5, y = 5 − x. Put into 2x − 3y = 4: 2x − 3(5 − x) = 4, so 5x − 15 = 4, 5x = 19, x = 19/5. Then y = 5 − 19/5 = 6/5.
  3. Both methods give the same pair.

Answerx = 19/5, y = 6/5.

(ii) 3x + 4y = 10 and 2x − 2y = 2

  1. Elimination: multiply 2x − 2y = 2 by 2 to get 4x − 4y = 4. Add it to 3x + 4y = 10: 7x = 14, so x = 2. Then 2(2) − 2y = 2 gives y = 1.
  2. Substitution: from 2x − 2y = 2, x = y + 1. Put into 3x + 4y = 10: 3(y + 1) + 4y = 10, so 7y = 7, y = 1. Then x = 1 + 1 = 2.
  3. Both methods give the same pair.

Answerx = 2, y = 1.

(iii) 3x − 5y − 4 = 0 and 9x = 2y + 7

  1. Write the pair as 3x − 5y = 4 and 9x − 2y = 7.
  2. Elimination: multiply 3x − 5y = 4 by 3 to get 9x − 15y = 12. Subtract this from 9x − 2y = 7: 13y = −5, so y = −5/13. Then 3x = 4 + 5y = 4 − 25/13 = 27/13, so x = 9/13.
  3. Substitution: from 3x − 5y = 4, x = (4 + 5y)/3, so 9x = 3(4 + 5y) = 12 + 15y. Put into 9x − 2y = 7: 12 + 15y − 2y = 7, so 13y = −5, y = −5/13. Then x = (4 − 25/13)/3 = (27/13)/3 = 9/13.
  4. Both methods give the same pair.

Answerx = 9/13, y = −5/13.

(iv) x/2 + 2y/3 = −1 and x − y/3 = 3

  1. Clear the fractions: multiply x/2 + 2y/3 = −1 by 6 to get 3x + 4y = −6, and multiply x − y/3 = 3 by 3 to get 3x − y = 9.
  2. Elimination: subtract 3x − y = 9 from 3x + 4y = −6: 5y = −15, so y = −3. Then 3x + 3 = 9, so x = 2.
  3. Substitution: from 3x − y = 9, y = 3x − 9. Put into 3x + 4y = −6: 3x + 4(3x − 9) = −6, so 15x − 36 = −6, 15x = 30, x = 2. Then y = 3(2) − 9 = −3.
  4. Both methods give the same pair.

Answerx = 2, y = −3.

Watch this explained “The four-step shape”, 8:06 into Elimination: scaling the equations so a variable cancels on addition

Question 2

“Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method” · p. 36

Open NCERT p. 36Matches NCERT’s answer

(i) add 1 to the numerator and subtract 1 from the denominator

  1. Let the fraction be x/y.
  2. Adding 1 to the numerator and subtracting 1 from the denominator gives 1: (x + 1)/(y − 1) = 1, so x + 1 = y − 1, that is x − y = −2.
  3. Adding 1 to the denominator only gives 1/2: x/(y + 1) = 1/2, so 2x = y + 1, that is 2x − y = 1.
  4. Subtract the first equation from the second: (2x − y) − (x − y) = 1 − (−2), so x = 3.
  5. Then from x − y = −2: y = 3 + 2 = 5.

AnswerThe fraction is 3/5.

(ii) Five years ago, Nuri was thrice as old as Sonu.

  1. Let Nuri's present age be x years and Sonu's be y years.
  2. Five years ago: x − 5 = 3(y − 5), so x − 3y = −10.
  3. Ten years later: x + 10 = 2(y + 10), so x − 2y = 10.
  4. Subtract the first equation from the second to remove x: (x − 2y) − (x − 3y) = 10 − (−10), so y = 20.
  5. Then x = 2(20) + 10 = 50.

AnswerNuri is 50 years old and Sonu is 20 years old.

(iii) The sum of the digits of a two-digit number is 9.

  1. Let the tens digit be x and the units digit be y. The number is 10x + y, and with the digits reversed it is 10y + x.
  2. The digits add to 9: x + y = 9.
  3. Nine times the number is twice the reversed number: 9(10x + y) = 2(10y + x), so 90x + 9y = 20y + 2x, that is 88x − 11y = 0, or 8x − y = 0.
  4. Add x + y = 9 and 8x − y = 0 to remove y: 9x = 9, so x = 1.
  5. Then y = 9 − 1 = 8, and the number is 10(1) + 8 = 18.
  6. Check: 9 × 18 = 162 and 2 × 81 = 162.

AnswerThe number is 18.

(iv) Meena went to a bank to withdraw ₹ 2000.

  1. Let the number of ₹50 notes be x and the number of ₹100 notes be y.
  2. Total notes: x + y = 25.
  3. Total value: 50x + 100y = 2000. Divide by 50: x + 2y = 40.
  4. Subtract x + y = 25 from x + 2y = 40: y = 15.
  5. Then x = 25 − 15 = 10.

AnswerMeena received 10 notes of ₹50 and 15 notes of ₹100.

(v) A lending library has a fixed charge for the first three days

  1. Let the fixed charge (for the first three days) be ₹x and the extra charge for each day after that be ₹y.
  2. Seven days means 4 extra days: x + 4y = 27.
  3. Five days means 2 extra days: x + 2y = 21.
  4. Subtract the second equation from the first: 2y = 6, so y = 3.
  5. Then x = 21 − 2 × 3 = 15.

AnswerThe fixed charge is ₹15 and the charge for each extra day is ₹3.

Watch this explained “Two ratios and one saving”, 2:16 into Elimination: scaling the equations so a variable cancels on addition

Every question here was solved twice, separately, by two different AI models, and each answer was put back into the question by a computer program to check it. Where the two disagreed, a stronger model solved it again and the computer check had to pass on its answer. A question about reasoning rather than a number is shown as “one way to think about it”, and anything not yet proven says so instead of guessing. Each answer links to the moment in the video that teaches it.

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