Pair of Linear Equations in Two Variables : Exercise - 3.2 (Mathematics NCERT Class 10th)



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Q.1      From the pair of linear equations in the following problems, and find their solution graphically.
(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boy and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens togeher cost Rs. 46. Find the cost of one pencil and that of one pen.
Sol.       (i) Let us denote the number of boys by x and the number of girls by y. Then, the equations formed are
              x + y = 10...(1)
              and, y = x + 4
               \Rightarrow – x + y = 4...(2)

              Let us draw the graphs of equations (1) and (2) by finding two solutions for each of these equations. The solutions of the equations are given in the table.
              For x + y = 10 
8
              For – x + y = 4
9
              Plotting the points and draw the lines passing through them to represent the equations as shown.

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               These two lines intersect at (3, 7). So, x = 3 and y = 7 is the required solution.
               Hence, the number of boys and girls are 3 and 7 respectively.
               Verification : Put x = 3 and y = 7 in (1) and (2), we find that both the equations are satisfied.

               (ii) Let us denote the cost of one pencil by Rs. x and one pen by Rs. y. Then, the equations formed are
               5x + 7y = 50 ...(1)
               and, 7x + 5y = 46 ...(2)
               Let us draw the graphs of equations (1) and (2) by finding two solutions for each of these equations.
               The solutions of the equations are given in the table.

               For 5x + 7y = 50
10
                For 7x + 5y = 46 
11
                Plotting the points and draw the lines passing through them to represent the equations as shown.

13
                These two lines intersect at (3, 5). So x = 3 and y = 5 is the required solution.
                Hence, the cost of one pencil is Rs. 3 and that of one pen is Rs. 5.
                Verification : Put x = 3 and y = 5 in (1) and (2), we find that both the equations are satisfied.


Q.2      On comparing the ratios {{{a_1}} \over {{a_2}}},{{{b_1}} \over {{b_2}}} and {{{c_1}} \over {{c_2}}} find out whether the representing the following pairs of linear equations intersect at a point, parallel or coincide.
            (i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0
            (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
            (iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0
Sol.        (i) The given pair of linear equations are
              5x – 4y + 8 = 0
              and, 7x + 6y – 9 = 0
              Here, {5 \over 7} \ne {{ - 4} \over 6}
               \Rightarrow (1) and (2) are intersecting lines.

              (ii) The given pair of linear equations are
              9x + 3y +12 = 0
              and, 18x + 6y + 24 = 0
              Here, {9 \over {18}} = {3 \over 6} = {{12} \over {24}} [Since, Each = {1 \over 2}]
              Therefore, (1) and (2) are coincident lines.

              (iii) The given pair of linear equations are
              6x – 3y + 10 = 0
              and, 2x – y + 9 = 0
              Here, {6 \over 2} = {{ - 3} \over { - 1}} \ne {{10} \over 9}
              Therefore, (1) and (2) are parallel lines


Q.3      On comparing the ratios {{{a_1}} \over {{a_2}}},{{{b_1}} \over {{b_2}}} and {{{c_1}} \over {{c_2}}}, find out whether the following pairs of linear equations are consistent, or inconsistent.
            (i) 3x + 2y = 5; 2x – 3y = 7
            (ii) 2x – 3y = 8; 4x – 6y = 9
            (iii) {3 \over 2}x + {5 \over 3}y = 7; 9x – 10y = 14
            (iv) 5x – 3y = 11; – 10x + 6y = – 22
            (v) {4 \over 3}x + 2y = 8; 2x + 3y = 12.
Sol.        (i) 3x + 2y = 5; 2x – 3y = 7
               3x + 2y – 5 = 0
               {{a_1} = 3},{{b_1} = 2},{{c_1} = - 5},{{a_2} = 2},{{b_2} = - 3},{{c_2} = - 7} and 2x – 3y – 7 = 0
              {{{a_1}} \over {{a_2}}} = {3 \over 2},{{{b_1}} \over {{b_2}}} = {2 \over { - 3}}
              Since, {3 \over 2} \ne {2 \over 3},{{{a_1}} \over {{a_2}}} \ne {{{b_1}} \over {{b_2}}}
              Therefore, The pair of linear equations is consistent.

              (ii) 2x – 3y = 8; 4x – 6y = 9; 2x – 3y – 8 = 0 and 4x – 6y – 3 = 0
              {{a_1} = 2},{{b_1} = - 3},{{c_1} = - 8}; {{a_2} = 4},{{b_2} = - 6},{{c_2} = - 9}
              {{{a_1}} \over {{a_2}}} = {2 \over 4} = {1 \over 2},{{{b_1}} \over {{b_2}}} = {{ - 3} \over { - 6}} = {1 \over 2},{{{c_1}} \over {{c_2}}} = {{ - 8} \over { - 9}} = {8 \over 9}
              Since, {1 \over 2} = {1 \over 2} \ne {8 \over 9},i.e.,{{{a_1}} \over {{a_2}}} = {{{b_1}} \over {{b_2}}} \ne {{{c_1}} \over {{c_2}}}
               Therefore, The pair of linear equations is inconsistent.

               (iii) {3 \over 2}x + {5 \over 3}y = 7; 9x – 10y = 14
               {a_1} = {3 \over 2},{b_1} = {5 \over 3},{c_1} = - 7;{a_2} = 9,{b_1} = - 10,{c_2} = - 14
               {{{a_1}} \over {{a_2}}} = {{{3 \over 2}} \over 9} = {1 \over 6},{{{b_1}} \over {{b_2}}} = {{{5 \over 3}} \over { - 10}} = - {1 \over 6}
               Since, {1 \over 6} \ne - {1 \over 6}, {{{a_1}} \over {{a_2}}} \ne {{{b_1}} \over {{b_2}}}
               Therefore, The pair of linear equations is consistent.

               (iv) 5x – 3y = 11; – 10x + 6y = – 22
               {a_1} = 5,{b_1} = - 3,{c_1} = - 11;{a_2} = - 10,{b_2} = 6,{c_2} = 22
               {{{a_1}} \over {{a_2}}} = {5 \over {10}} = - {1 \over 2},{{{b_1}} \over {{b_2}}} = {{ - 3} \over 6}= - {1 \over 2},{{{c_1}} \over {{c_2}}} = {{ - 11} \over {22}} = - {1 \over 2}
                Since, - {1 \over 2} = - {1 \over 2} = - {1 \over 2},i.e.,{{{a_1}} \over {{a_2}}} = {{{b_1}} \over {{b_2}}} = {{{c_1}} \over {{c_2}}}

               Therefore, The pair of linear equation is consistent.

              (v) {4 \over 3}x + 2y = 8;2x + 3y = 12
              {a_1} = {4 \over 3},{b_1} = 2,{c_1} = - 8;{a_2} = 2,{b_2} = 3,{c_2} = - 12
              {{{a_1}} \over {{a_2}}} = {{{4 \over 3}} \over 2} = {2 \over 3},{{{b_1}} \over {{b_2}}} = {2 \over 3},{{{c_1}} \over {{c_2}}} = {{ - 8} \over { - 12}} = {2 \over 3}
              Since, {2 \over 3} = {2 \over 3} = {2 \over 3},i.e.,{{{a_1}} \over {{a_2}}} = {{{b_1}} \over {{b_2}}} = {{{c_1}} \over {{c_2}}}
              Therefore, The pair of linear equation is consistent.


Q.4      Which of the following pairs of linear equations are consistent ? Obtain the solution in such cases graphically.
           (i) x + y = 5, 2x + 2y = 10
           (ii) x – y = 8, 3x – 3y = 16
           (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
           (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Sol.       (i) Graph of x + y = 5 :
             We have, x + y = 5
              \Rightarrow y = 5 – x
             When x = 0, y = 5; when x = 5, y = 0
             Thus, we have the following table :

14
               Plotting the points A(0, 5) and (5, 0) on the graph paper. Join A and B and extend it on both sides to obtain he graph of  x + y = 5.

201               Graph of 2x +2y = 10 :
               We have 2x + 2y = 10
                \Rightarrow 2y = 10 – 2x
                \Rightarrow y = 5 – x

               When x = 1, y = 5 – 1 = 4; when x = 2, y = 5 – 2 = 3
               Thus, we have the following table :
16
             Plotting the points C(1, 4) and D(2, 3) on the graph paper and drawing a line passing through these points on the same graph paper, we obtain the graph of 2x + 2y = 10
             We find that C and D both lie on the graph of x + y = 5. Thus, the graphs of the two equations are coincident .              Consequently, every solution of one equation is a solution the other.
             Hence, the system of equations has infinitely many solutions, i.e., consistent.

             (ii) Graph of x – y = 8 :
             We have, x = 0, y = – 8; when x = 8, y = 0
             Thus, we have the following table :

102
Plotting the points A(0, – 8) and B(8, 0) on a graph paper. Join A and B and extend it on both sides to obtain the graph of x – y = 8 as shown.

             graph of 3x – 3y = 16:
             We have, 3x – 3y = 16
              \Rightarrow 3y = 3x – 16 
              \Rightarrow y = {{3x - 16} \over 3}

             When x = 0, y = {{ - 16} \over 3} = - 5{1 \over 3};
             When x = {{16} \over 3} = 5{1 \over 3}, y = 0
             Thus, we have the following table :

18
             Plot the points C\left( {0,{{ - 16} \over 3}} \right) and D\left( {{{16} \over 3},0} \right) on the same graph paper. Join C and D and extend it on both sides to obtain the graph of 3x – 3y = 16 as shown.

19
               We find the graphs of x – y = 8 and 3x – 3y = 16 are parallel. So, the two lines have no common point. Hence, the given equations has no solution, i.e., inconsistent.

               (iii) Graph of 2x + y – 6 = 0 : 
               We have, 2x + y – 6 = 0
                \Rightarrow y = 6 – 2x

               When x = 0, y = 6 – 0 = 6; when x = 3, y = 6 – 6 = 0
               Thus, we have the following table :

20
                 Plot the points A(0, 6) and B(3, 0) on a graph paper. Join A and B and extend it on both sides to obtain graph of 2x + y – 6 = 0 as shown.
                Graph of 4x – 2y – 4 = 0 :
                We have,
                4x – 2y – 4 = 0
                 \Rightarrow 2y = 4x – 4
                 \Rightarrow y = 2x – 2

                When x = 0 , y = – 2; when x = 1, y = 0
                Thus, we have the following table :

21
              Plotting the points C(0, –2) and D(2, 0) on the same graph and drawing a line joining them as shown.
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             Clearly, the two lines intersect at point P(2, 2).
             Hence, x = 2, y = 2 is the solution of the given equations, i.e., consistent.
             Verification : Putting x = 2, y = 2 in the given equations, we find that both the equations are satisfied.

            (iv) Graph of 2x – 2y – 2 = 0;
            We have,
            2x – 2y – 2 = 0
             \Rightarrow 2y = 2x – 2
             \Rightarrow y = x – 2
            When x = 2, y = 0; when x = 0, y = – 2
            Thus, we have the following table :

22
            Plot the points A(2, 0) and B(0, –2) on a graph paper. Join A and B and extend it on both sides to obtain the graph of 2x –2y – 2 = 0.
           Graph of 4x – 4y – 5 = 0 :
           We have,
           4x – 4y – 5 = 0
            \Rightarrow 4y = 5 – 4x
            \Rightarrow y = {{5 - 4x} \over 4}

           When x = 0, y =  - {5 \over 4}; when x = {5 \over 4}, y = 0
           Thus, we have the following table :
23
          Plot the points C\left( {0,{{ - 5} \over 4}} \right) and D\left( {{5 \over 4},0} \right) on the same
graph paper. Join C and D and extend it on both sides to obtain the graph of 4x – 4y – 5 = 0 as shown.

106
             We find the graphs of these equations are parallel lines. So, the two lines have no common point. Hence, the given system of equations has no solution, i.e., inconsistent.

Q.5      Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the graden.
Sol.        Let the length of the garden be x m and its width be y m.
              Then, perimeter = 2(Length + Width) = 2(x + y)
              But perimeter = 36 [Given]
              Therefore, 2(x + y) = 36
               \Rightarrow x + y = {{36} \over 2} = 18

              Also, x = y + 4
              Therefore, y + 4 + y = 18
               \Rightarrow 2y = 18 – 4 = 14

               \Rightarrow y = {{14} \over 2} = 7
              Therefore, x = y + 4 = 7 + 4 = 11
              Hence, Length = 11m and width = 7m.


Q.6      Given the linear equation 2x + 3y – 8 = 0, write other linear equation in two variables such that the geometrical representation of the pair so formed
           (i) intersecting lines
           (ii) parallel lines
           (iii) coincident lines
Sol.       Given linear equation is 2x + 3y – 8 = 0 ...(1)
             (i) For intersecting lines, we know that
             {{{a_1}} \over {{a_2}}} \ne {{{b_1}} \over {{b_2}}}
             Any line intersecting line may be taken as
             5x + 2y – 9 = 0

            (ii) For parallel lines, {{{a_1}} \over {{a_2}}} = {{{b_1}} \over {{b_2}}} \ne {{{c_1}} \over {{c_2}}}
            Therefore, Any line parallel to (1) may be taken as
            4x + 6y –3 = 0

           (iii) For coincident lines, {{{a_1}} \over {{a_2}}} = {{{b_1}} \over {{b_2}}} = {{{c_1}} \over {{c_2}}}
           Any line coincident to (1) may be taken as
           6x + 9x – 24 = 0          


Q.7      Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
Sol.       For the graph of x – y + 1 = 0.
             We have,
             x – y + 1 = 0
              \Rightarrow y = x + 1
             When x = 0, y = 1; when x = – 1, y = 0
             Thus, we have the following table :

252525
             Plot the points A(0, 1) and B(–1, 0) on a graph paper. Join A and B and extend it on both sides to obtain  the graph of x – y + 1 = 0.
             For the graph of 3x + 2y – 12 = 0 :
             We have,
             3x + 2y – 12 = 0
              \Rightarrow 2y = 12 – 3x
              \Rightarrow y = {{12 - 3x} \over 2}

             When x = 4, y = {{12 - 12} \over 2} = 0
             When x = 0, y = {{12} \over 2} = 6
             Thus, we have the following table :

26
              Plot the points C(4, 0) and D(0, 6) on the same graph paper and draw a line passing through these two points to obtain the graph of equation 3x + 2y – 12 = 0.

115
               Clearly, we obtain a \Delta PBC formed by the given lines and the x - axis. The co-ordinates of the vertices are P(2, 3), B(–1, 0) and C(4, 0).

 



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