Algebra · Textbook solutions

Linear Equations in Two Variables

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 80 questions

Simultaneous linear equations

3 q

Solved Examples

Worked · 3
  1. Simultaneous linear equations SolvedEx.1
    Solve the following simultaneous equations. 5x−3y=85x - 3y = 8; 3x+y=23x + y = 2
  2. Simultaneous linear equations SolvedEx.2
    Solve : 3x+2y=293x + 2y = 29; 5x−y=185x - y = 18
  3. Simultaneous linear equations SolvedEx.3
    Solve : 15x+17y=2115x + 17y = 21; 17x+15y=1117x + 15y = 11

Practice set 1.1

9 q
  1. Ex 1.1 Q.1
    Complete the following activity to solve the simultaneous equations. 5x+3y=95x + 3y = 9 -----(I) 2x−3y=122x - 3y = 12 ----- (II) Add equations (I) and (II) and find the value of xx. Then place that value of xx in equation (I) to find yy, and write the solution (x,y)(x, y).
  2. Solve the following simultaneous equations.
    Ex 1.1 Q.2(1)
    3a+5b=263a + 5b = 26; a+5b=22a + 5b = 22
  3. Ex 1.1 Q.2(2)
    x+7y=10x + 7y = 10; 3x−2y=73x - 2y = 7
  4. Ex 1.1 Q.2(3)
    2x−3y=92x - 3y = 9; 2x+y=132x + y = 13
  5. Ex 1.1 Q.2(4)
    5m−3n=195m - 3n = 19; m−6n=−7m - 6n = -7
  6. Ex 1.1 Q.2(5)
    5x+2y=−35x + 2y = -3; x+5y=4x + 5y = 4
  7. Ex 1.1 Q.2(6)
    13x+y=103\dfrac{1}{3}x + y = \dfrac{10}{3}; 2x+14y=1142x + \dfrac{1}{4}y = \dfrac{11}{4}
  8. Ex 1.1 Q.2(7)
    99x+101y=49999x + 101y = 499; 101x+99y=501101x + 99y = 501
  9. Ex 1.1 Q.2(8)
    49x−57y=17249x - 57y = 172; 57x−49y=25257x - 49y = 252

Practice set 1.2

8 q
  1. Complete the following table to draw graph of the equations.
    Ex 1.2 Q.1(I)
    (I) x+y=3x + y = 3
    xx3
    yy53
    (x,y)(x, y)(3, 0)(0, 3)
  2. Ex 1.2 Q.1(II)
    (II) x−y=4x - y = 4
    xx−1-10
    yy0−4-4
    (x,y)(x, y)(0, −4-4)
  3. Solve the following simultaneous equations graphically.
    Ex 1.2 Q.2(1)
    x+y=6x + y = 6 ; x−y=4x - y = 4
  4. Ex 1.2 Q.2(2)
    x+y=5x + y = 5 ; x−y=3x - y = 3
  5. Ex 1.2 Q.2(3)
    x+y=0x + y = 0 ; 2x−y=92x - y = 9
  6. Ex 1.2 Q.2(4)
    3x−y=23x - y = 2 ; 2x−y=32x - y = 3
  7. Ex 1.2 Q.2(5)
    3x−4y=−73x - 4y = -7 ; 5x−2y=05x - 2y = 0
  8. Ex 1.2 Q.2(6)
    2x−3y=42x - 3y = 4 ; 3y−x=43y - x = 4

Determinant

3 q

Solved Examples

Worked · 3
  1. Find the values of the following determinants.
    Determinant SolvedEx.1(1)
    A=∣5379∣\mathrm{A} = \begin{vmatrix} 5 & 3 \\ 7 & 9 \end{vmatrix}
  2. Determinant SolvedEx.1(2)
    N=∣−8−324∣\mathrm{N} = \begin{vmatrix} -8 & -3 \\ 2 & 4 \end{vmatrix}
  3. Determinant SolvedEx.1(3)
    B=∣239233∣\mathrm{B} = \begin{vmatrix} 2\sqrt{3} & 9 \\ 2 & 3\sqrt{3} \end{vmatrix}

Determinant method (Cramer's Rule)

1 q

Solved Examples

Worked · 1
  1. Determinant method (Cramer's Rule) SolvedEx.1
    Solve the following simultaneous equations using Cramer's Rule. 5x+3y=−115x + 3y = -11 ; 2x+4y=−102x + 4y = -10

Practice set 1.3

10 q
  1. Ex 1.3 Q.1
    Fill in the blanks with correct number ∣3245∣=3×□−□×4=□−8=□\begin{vmatrix} 3 & 2 \\ 4 & 5 \end{vmatrix} = 3 \times \square - \square \times 4 = \square - 8 = \square
  2. Find the values of following determinants.
    Ex 1.3 Q.2(1)
    ∣−1724∣\begin{vmatrix} -1 & 7 \\ 2 & 4 \end{vmatrix}
  3. Ex 1.3 Q.2(2)
    ∣53−70∣\begin{vmatrix} 5 & 3 \\ -7 & 0 \end{vmatrix}
  4. Ex 1.3 Q.2(3)
    ∣73533212∣\begin{vmatrix} \dfrac{7}{3} & \dfrac{5}{3} \\ \dfrac{3}{2} & \dfrac{1}{2} \end{vmatrix}
  5. Solve the following simultaneous equations using Cramer's rule.
    Ex 1.3 Q.3(1)
    3x−4y=103x - 4y = 10 ; 4x+3y=54x + 3y = 5
  6. Ex 1.3 Q.3(2)
    4x+3y−4=04x + 3y - 4 = 0 ; 6x=8−5y6x = 8 - 5y
  7. Ex 1.3 Q.3(3)
    x+2y=−1x + 2y = -1 ; 2x−3y=122x - 3y = 12
  8. Ex 1.3 Q.3(4)
    6x−4y=−126x - 4y = -12 ; 8x−3y=−28x - 3y = -2
  9. Ex 1.3 Q.3(5)
    4m+6n=544m + 6n = 54 ; 3m+2n=283m + 2n = 28
  10. Ex 1.3 Q.3(6)
    2x+3y=22x + 3y = 2 ; x−y2=12x - \dfrac{y}{2} = \dfrac{1}{2}

Equations reducible to a pair of linear equations in two variables

2 q

Solved Examples

Worked · 2
  1. Equations reducible SolvedEx.1
    Solve: 4x+5y=7\dfrac{4}{x} + \dfrac{5}{y} = 7; 3x+4y=5\dfrac{3}{x} + \dfrac{4}{y} = 5
  2. Equations reducible SolvedEx.2
    Solve: 4x−y+1x+y=3\dfrac{4}{x - y} + \dfrac{1}{x + y} = 3 ; 2x−y−3x+y=5\dfrac{2}{x - y} - \dfrac{3}{x + y} = 5

Practice set 1.4

4 q
  1. Solve the following simultaneous equations.
    Ex 1.4 Q.1(1)
    2x−3y=15\dfrac{2}{x} - \dfrac{3}{y} = 15 ; 8x+5y=77\dfrac{8}{x} + \dfrac{5}{y} = 77
  2. Ex 1.4 Q.1(2)
    10x+y+2x−y=4\dfrac{10}{x + y} + \dfrac{2}{x - y} = 4 ; 15x+y−5x−y=−2\dfrac{15}{x + y} - \dfrac{5}{x - y} = -2
  3. Ex 1.4 Q.1(3)
    27x−2+31y+3=85\dfrac{27}{x - 2} + \dfrac{31}{y + 3} = 85 ; 31x−2+27y+3=89\dfrac{31}{x - 2} + \dfrac{27}{y + 3} = 89
  4. Ex 1.4 Q.1(4)
    13x+y+13x−y=34\dfrac{1}{3x + y} + \dfrac{1}{3x - y} = \dfrac{3}{4} ; 12(3x+y)−12(3x−y)=−18\dfrac{1}{2(3x + y)} - \dfrac{1}{2(3x - y)} = -\dfrac{1}{8}

Application of Simultaneous equations

5 q

Solved Examples

Worked · 5
  1. Application of Simultaneous equations SolvedEx.1
    The perimeter of a rectangle is 40 cm. The length of the rectangle is more than double its breadth by 2. Find length and breadth.
  2. Application of Simultaneous equations SolvedEx.2
    Sale ! Sale !! Sale !!! only for 2 days I have some analogue wrist watches and some digital wrist watches. I am going to sell them at a discount. Sale of 1st day : Analogue watch = 11, Digital watch = 6, Received amount = ₹ 4330 Sale of the 2nd day : Analogue watch = 22, Digital watch = 5, Received amount = ₹ 7330 Find selling price of wrist watch of each type.
  3. Application of Simultaneous equations SolvedEx.3
    A boat travels 16 km upstream and 24 km downstream in 6 hours. The same boat travels 36 km upstream and 48 km downstream in 13 hours. Find the speed of water current and speed of boat in still water.
  4. Application of Simultaneous equations SolvedEx.4
    A certain amount is equally distributed among certain number of students. Each would get ₹ 2 less if 10 students were more and each would get ₹ 6 more if 15 students were less. Find the number of students and the amount distributed.
  5. Application of Simultaneous equations SolvedEx.5
    A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.

Practice set 1.5

5 q
  1. Ex 1.5 Q.2
    Complete the following. A rectangle has its four sides labelled as follows: the top side is 2x+y+82x + y + 8, the bottom side is 4x−y4x - y, the right side is x+4x + 4 and the left side is 2y2y. Find the values of xx and yy. Find my perimeter and area.
  2. Ex 1.5 Q.3
    The sum of father's age and twice the age of his son is 70. If we double the age of the father and add it to the age of his son the sum is 95. Find their present ages.
  3. Ex 1.5 Q.4
    The denominator of a fraction is 4 more than twice its numerator. Denominator becomes 12 times the numerator, if both the numerator and the denominator are reduced by 6. Find the fraction.
  4. Ex 1.5 Q.5
    Two types of boxes A, B are to be placed in a truck having capacity of 10 tons. When 150 boxes of type A and 100 boxes of type B are loaded in the truck, it weighes 10 tons. But when 260 boxes of type A are loaded in the truck, it can still accommodate 40 boxes of type B, so that it is fully loaded. Find the weight of each type of box.
  5. Ex 1.5 Q.6
    Out of 1900 km, Vishal travelled some distance by bus and some by aeroplane. Bus travels with average speed 60 km/hr and the average speed of aeroplane is 700 km/hr. It takes 5 hours to complete the journey. Find the distance, Vishal travelled by bus.

Problem set 1

30 q
  1. Choose correct alternative for each of the following questions
    PS1 Q.1(1)
    To draw graph of 4x+5y=194x + 5y = 19, Find yy when x=1x = 1.
    1. A.
      4
    2. B.
      3
    3. C.
      2
    4. D.
      −3-3
  2. PS1 Q.1(2)
    For simultaneous equations in variables xx and yy, Dx=49\mathrm{D}_x = 49, Dy=−63\mathrm{D}_y = -63, D=7\mathrm{D} = 7 then what is xx ?
    1. A.
      7
    2. B.
      −7-7
    3. C.
      17\dfrac{1}{7}
    4. D.
      −17\dfrac{-1}{7}
  3. PS1 Q.1(3)
    Find the value of ∣53−7−4∣\begin{vmatrix} 5 & 3 \\ -7 & -4 \end{vmatrix}
    1. A.
      −1-1
    2. B.
      −41-41
    3. C.
      41
    4. D.
      1
  4. PS1 Q.1(4)
    To solve x+y=3x + y = 3 ; 3x−2y−4=03x - 2y - 4 = 0 by determinant method find D.
    1. A.
      5
    2. B.
      1
    3. C.
      −5-5
    4. D.
      −1-1
  5. PS1 Q.1(5)
    ax+by=cax + by = c and mx+ny=dmx + ny = d and an≠bman \neq bm then these simultaneous equations have -
    1. A.
      Only one common solution.
    2. B.
      No solution.
    3. C.
      Infinite number of solutions.
    4. D.
      Only two solutions.
  6. PS1 Q.2
    Complete the following table to draw the graph of 2x−6y=32x - 6y = 3
    xx−5-5
    yy0
    (x,y)(x, y)
  7. Solve the following simultaneous equations graphically.
    PS1 Q.3(1)
    2x+3y=122x + 3y = 12 ; x−y=1x - y = 1
  8. PS1 Q.3(2)
    x−3y=1x - 3y = 1 ; 3x−2y+4=03x - 2y + 4 = 0
  9. PS1 Q.3(3)
    5x−6y+30=05x - 6y + 30 = 0 ; 5x+4y−20=05x + 4y - 20 = 0
  10. PS1 Q.3(4)
    3x−y−2=03x - y - 2 = 0 ; 2x+y=82x + y = 8
  11. PS1 Q.3(5)
    3x+y=103x + y = 10 ; x−y=2x - y = 2
  12. Find the values of each of the following determinants.
    PS1 Q.4(1)
    ∣4327∣\begin{vmatrix} 4 & 3 \\ 2 & 7 \end{vmatrix}
  13. PS1 Q.4(2)
    ∣5−2−31∣\begin{vmatrix} 5 & -2 \\ -3 & 1 \end{vmatrix}
  14. PS1 Q.4(3)
    ∣3−114∣\begin{vmatrix} 3 & -1 \\ 1 & 4 \end{vmatrix}
  15. Solve the following equations by Cramer's method.
    PS1 Q.5(1)
    6x−3y=−106x - 3y = -10 ; 3x+5y−8=03x + 5y - 8 = 0
  16. PS1 Q.5(2)
    4m−2n=−44m - 2n = -4 ; 4m+3n=164m + 3n = 16
  17. PS1 Q.5(3)
    3x−2y=523x - 2y = \dfrac{5}{2} ; 13x+3y=−43\dfrac{1}{3}x + 3y = -\dfrac{4}{3}
  18. PS1 Q.5(4)
    7x+3y=157x + 3y = 15 ; 12y−5x=3912y - 5x = 39
  19. PS1 Q.5(5)
    x+y−82=x+2y−143=3x−y4\dfrac{x + y - 8}{2} = \dfrac{x + 2y - 14}{3} = \dfrac{3x - y}{4}
  20. Solve the following simultaneous equations.
    PS1 Q.6(1)
    2x+23y=16\dfrac{2}{x} + \dfrac{2}{3y} = \dfrac{1}{6} ; 3x+2y=0\dfrac{3}{x} + \dfrac{2}{y} = 0
  21. PS1 Q.6(2)
    72x+1+13y+2=27\dfrac{7}{2x + 1} + \dfrac{13}{y + 2} = 27 ; 132x+1+7y+2=33\dfrac{13}{2x + 1} + \dfrac{7}{y + 2} = 33
  22. PS1 Q.6(3)
    148x+231y=527xy\dfrac{148}{x} + \dfrac{231}{y} = \dfrac{527}{xy} ; 231x+148y=610xy\dfrac{231}{x} + \dfrac{148}{y} = \dfrac{610}{xy}
  23. PS1 Q.6(4)
    7x−2yxy=5\dfrac{7x - 2y}{xy} = 5 ; 8x+7yxy=15\dfrac{8x + 7y}{xy} = 15
  24. PS1 Q.6(5)
    12(3x+4y)+15(2x−3y)=14\dfrac{1}{2(3x + 4y)} + \dfrac{1}{5(2x - 3y)} = \dfrac{1}{4} ; 5(3x+4y)−2(2x−3y)=−32\dfrac{5}{(3x + 4y)} - \dfrac{2}{(2x - 3y)} = -\dfrac{3}{2}
  25. Solve the following word problems.
    PS1 Q.7(1)
    A two digit number and the number with digits interchanged add up to 143. In the given number the digit in unit's place is 3 more than the digit in the ten's place. Find the original number. Let the digit in unit's place is xx and that in the ten's place is yy. Frame the two equations from the given conditions and solve them to find the original number.
  26. PS1 Q.7(2)
    Kantabai bought 1121\dfrac{1}{2} kg tea and 5 kg sugar from a shop. She paid ₹ 50 as return fare for rickshaw. Total expense was ₹ 700. Then she realised that by ordering online the goods can be bought with free home delivery at the same price. So next month she placed the order online for 2 kg tea and 7 kg sugar. She paid ₹ 880 for that. Find the rate of sugar and tea per kg.
  27. PS1 Q.7(3)
    To find number of notes that Anushka had, complete the following activity. Suppose that Anushka had xx notes of ₹ 100 and yy notes of ₹ 50 each. Anushka got ₹ 2500/- from Anand as denominations mentioned above. ∴\therefore write equation I. If Anand would have given her the amount by interchanging number of notes, Anushka would have received ₹ 500 less than the previous amount. ∴\therefore write equation II. ∴\therefore Find the number of notes of each denomination.
  28. PS1 Q.7(4)
    Sum of the present ages of Manish and Savita is 31. Manish's age 3 years ago was 4 times the age of Savita. Find their present ages.
  29. PS1 Q.7(5)
    In a factory the ratio of salary of skilled and unskilled workers is 5 : 3. Total salary of one day of both of them is ₹ 720. Find daily wages of skilled and unskilled workers.
  30. PS1 Q.7(6)
    Places A and B are 30 km apart and they are on a straight road. Hamid travels from A to B on bike. At the same time Joseph starts from B on bike, travels towards A. They meet each other after 20 minutes. If Joseph would have started from B at the same time but in the opposite direction (instead of towards A) Hamid would have caught him after 3 hours. Find the speed of Hamid and Joseph.