Mathematics · Textbook solutions

Linear Equations in Two Variables

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 50 questions

Elimination Method

4 q

Solved Examples

Worked · 4
  1. Elimination SolvedEx.1
    Solve x+y=14x + y = 14 and x−y=2x - y = 2.
  2. Elimination SolvedEx.2
    Sum of the ages of mother and son is 45 years. If son's age is subtracted from twice of mother's age then we get answer 54. Find the ages of mother and son.
  3. Elimination SolvedEx.3
    Solve the following equations : 3x+y=53x + y = 5 and 2x+3y=12x + 3y = 1.
  4. Elimination SolvedEx.4
    Solve the following simultaneous equations : 3x−4y−15=03x - 4y - 15 = 0 and y+x+2=0y + x + 2 = 0.

Substitution Method

2 q

Solved Examples

Worked · 2
  1. Substitution SolvedEx.1
    Solve 8x+3y=118x + 3y = 11 ; 3x−y=23x - y = 2.
  2. Substitution SolvedEx.2
    Solve. 3x−4y=163x - 4y = 16 ; 2x−3y=102x - 3y = 10.

Practice Set 5.1

8 q
  1. Ex 5.1 Q1
    By using variables xx and yy form any five linear equations in two variables.
  2. Ex 5.1 Q2
    Write five solutions of the equation x+y=7x + y = 7.
  3. Solve the following sets of simultaneous equations.
    Ex 5.1 Q3(i)
    x+y=4x + y = 4 ; 2x−5y=12x - 5y = 1
  4. Ex 5.1 Q3(ii)
    2x+y=52x + y = 5 ; 3x−y=53x - y = 5
  5. Ex 5.1 Q3(iii)
    3x−5y=163x - 5y = 16 ; x−3y=8x - 3y = 8
  6. Ex 5.1 Q3(iv)
    2y−x=02y - x = 0 ; 10x+15y=10510x + 15y = 105
  7. Ex 5.1 Q3(v)
    2x+3y+4=02x + 3y + 4 = 0 ; x−5y=11x - 5y = 11
  8. Ex 5.1 Q3(vi)
    2x−7y=72x - 7y = 7 ; 3x+y=223x + y = 22

Word Problems on Simultaneous Equations

4 q

Solved Examples

Worked · 4
  1. 5.2 SolvedEx.1
    Sum of two numbers is 103. If greater number is divided by smaller number then the quotient is 2 and the remainder is 19. Then find the numbers.
  2. 5.2 SolvedEx.2
    Salil's age is 23 years more than half of the Sangram's age. Five years ago, the sum of their ages was 55 years. Find their present ages.
  3. 5.2 SolvedEx.3
    A two digit number is 4 times the sum of its digits. If we interchange the digits, the number obtained is 9 less than 4 times the original number. Then find the number.
  4. 5.2 SolvedEx.4
    The population of a certain town was 50,000. In a year, male population was increased by 5% and female population was increased by 3%. Now the population became 52020. Then what was the number of males and females in the previous year?

Practice Set 5.2

10 q
  1. Ex 5.2 Q1
    In an envelope there are some 5 rupee notes and some 10 rupee notes. Total amount of these notes together is 350 rupees. Number of 5 rupee notes are less by 10 than number of 10 rupee notes. Then find the number of 5 rupee and 10 rupee notes.
  2. Ex 5.2 Q2
    The denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator and denominator respectively, the ratio of numerator to denominator is 3 : 5. Find the fraction.
  3. Ex 5.2 Q3
    The sum of ages of Priyanka and Deepika is 34 years. Priyanka is elder to Deepika by 6 years. Then find their today's ages.
  4. Ex 5.2 Q4
    The total number of lions and peacocks in a certain zoo is 50. The total number of their legs is 140. Then find the number of lions and peacocks in the zoo.
  5. Ex 5.2 Q5
    Sanjay gets fixed monthly income. Every year there is a certain increment in his salary. After 4 years, his monthly salary was Rs. 4500 and after 10 years his monthly salary became 5400 rupees, then find his original salary and yearly increment.
  6. Ex 5.2 Q6
    The price of 3 chairs and 2 tables is 4500 rupees and price of 5 chairs and 3 tables is 7000 rupees, then find the total price of 2 chairs and 2 tables.
  7. Ex 5.2 Q7
    The sum of the digits in a two-digits number is 9. The number obtained by interchanging the digits exceeds the original number by 27. Find the two-digit number.
  8. Ex 5.2 Q8
    In △ABC\triangle ABC, the measure of angle A is equal to the sum of the measures of ∠B\angle B and ∠C\angle C. Also the ratio of measures of ∠B\angle B and ∠C\angle C is 4 : 5. Then find the measures of angles of the triangle.
  9. Ex 5.2 Q9
    Divide a rope of length 560 cm into 2 parts such that twice the length of the smaller part is equal to 13\dfrac{1}{3} of the larger part. Then find the length of the larger part.
  10. Ex 5.2 Q10
    In a competitive examination, there were 60 questions. The correct answer would carry 2 marks, and for incorrect answer 1 mark would be subtracted. Yashwant had attempted all the questions and he got total 90 marks. Then how many questions he got wrong ?

Problem Set 5

22 q
  1. Choose the correct alternative answers for the following questions.
    Prob Q1(i)
    If 3x+5y=93x + 5y = 9 and 5x+3y=75x + 3y = 7 then What is the value of x+yx + y ?
    1. A.
      2
    2. B.
      16
    3. C.
      9
    4. D.
      7
  2. Prob Q1(ii)
    'When 5 is subtracted from length and breadth of the rectangle, the perimeter becomes 26.' What is the mathematical form of the statement ?
    1. A.
      x−y=8x - y = 8
    2. B.
      x+y=8x + y = 8
    3. C.
      x+y=23x + y = 23
    4. D.
      2x+y=212x + y = 21
  3. Prob Q1(iii)
    Ajay is younger than Vijay by 5 years. Sum of their ages is 25 years. What is Ajay's age ?
    1. A.
      20
    2. B.
      15
    3. C.
      10
    4. D.
      5
  4. Solve the following simultaneous equations.
    Prob Q2(i)
    2x+y=52x + y = 5 ; 3x−y=53x - y = 5
  5. Prob Q2(ii)
    x−2y=−1x - 2y = -1 ; 2x−y=72x - y = 7
  6. Prob Q2(iii)
    x+y=11x + y = 11 ; 2x−3y=72x - 3y = 7
  7. Prob Q2(iv)
    2x+y=−22x + y = -2 ; 3x−y=73x - y = 7
  8. Prob Q2(v)
    2x−y=52x - y = 5 ; 3x+2y=113x + 2y = 11
  9. Prob Q2(vi)
    x−2y=−2x - 2y = -2 ; x+2y=10x + 2y = 10
  10. By equating coefficients of variables, solve the following equations.
    Prob Q3(i)
    3x−4y=73x - 4y = 7 ; 5x+2y=35x + 2y = 3
  11. Prob Q3(ii)
    5x+7y=175x + 7y = 17 ; 3x−2y=43x - 2y = 4
  12. Prob Q3(iii)
    x−2y=−10x - 2y = -10 ; 3x−5y=−123x - 5y = -12
  13. Prob Q3(iv)
    4x+y=344x + y = 34 ; x+4y=16x + 4y = 16
  14. Solve the following simultaneous equations.
    Prob Q4(i)
    x3+y4=4\dfrac{x}{3} + \dfrac{y}{4} = 4 ; x2−y4=1\dfrac{x}{2} - \dfrac{y}{4} = 1
  15. Prob Q4(ii)
    x3+5y=13\dfrac{x}{3} + 5y = 13 ; 2x+y2=192x + \dfrac{y}{2} = 19
  16. Prob Q4(iii)
    2x+3y=13\dfrac{2}{x} + \dfrac{3}{y} = 13 ; 5x−4y=−2\dfrac{5}{x} - \dfrac{4}{y} = -2
  17. Prob Q5
    A two digit number is 3 more than 4 times the sum of its digits. If 18 is added to this number, the sum is equal to the number obtained by interchanging the digits. Find the number.
  18. Prob Q6
    The total cost of 8 books and 5 pens is 420 rupees and the total cost of 5 books and 8 pens is 321 rupees. Find the cost of 1 book and 2 pens.
  19. Prob Q7
    The ratio of incomes of two persons is 9 : 7. The ratio of their expenses is 4 : 3. Every person saves rupees 200, find the income of each.
  20. Prob Q8
    If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square units. Then find the length and breadth of the rectangle.
  21. Prob Q9
    The distance between two places A and B on road is 70 kilometers. A car starts from A and the other from B. If they travel in the same direction, they will meet after 7 hours. If they travel towards each other they will meet after 1 hour, then find their speeds.
  22. Prob Q10
    The sum of a two digit number and the number obtained by interchanging its digits is 99. Find the number.