Q. 415.0( 2 Votes )

Find the solutions of the form x = a, y = 0 and x = 0, y = b for each of the following pairs of equations. 
         Do they have any common such solution?
         (i)   3x + 2y = 6 and 5x - 2y =10
         (ii)  5x + 3y = 15 and 5x + 2y =10
         (iii) 9x + 7y = 63 and x - y =10

Answer :

(i) 3x+2y = 6 and 5x-2y = 10
    Put y = 0, 3x+ 2൰ = 6 
     or 3x = 6
         ∴ x = 2
 Put x = 0 in 3x+2y = 6
                   3 × 0+2 × y = 6
                                   2y = 6
                                   y = 3
∴ The solutions of the equation 3x+2y = 6 is x = 2, y = 0 and x = 0, y = 3.
     5x-2y = 10
   Take y = 0, 5x-2 × 0 = 10
      or 5x = 10
          ∴ x = 2.
     Put x = 0 in 5x-2y = 10
    5 × 0-2y = 10,
         ⇒ -2y = 10
            ∴ y = -5,
     The solutions of the equation 5x-2y = 10 are x = 2, y = 0; x = 0, y = -5.
 ∴ The solution x = 2, y = 0 is common to both the equations.
(ii) 5x+3y = 15 and 5x+2y = 10
        5x+3y = 15
     Put y = 0 in 5x+3 × 0 = 15
     or 5x = 15
           ∴ x = 3
     Put x = 0 in 5x+3y = 15
     5  × 0+3y = 15
             ⇒ 3y = 15
               ∴ y = 5
The solutions of the equation 5x+3y = 15 are x = 3, y = 0 and x = 0, y = 5.
∴The solution x = 0, y = 5 is common to both the equations.
Put y = 0 in 5x+2y = 10
     5x+2 × 0 = 10
             ⇒ 5x = 10
            ∴  x = 2
     Put x = 0 in 5x+2y = 10
     5 × 0+2y = 10
                2y = 10
                   y = 5
∴ The solutions of the equation 5x+2y = 10 are x = 2, y = 0 and x = 0, y = 5.
(iii) 9x+7y = 63 and x - y = 10
      Put y = 0 in 9x+7y = 63
                   9x+7 × 0 = 63
                           ⇒  9x = 63
                               ∴ x = 7
      Put x = 0 in 9x+7y = 63
                   9 × 0+7y = 63
                          ⇒ 7y = 63
                               ∴ y = 9
      The solutions of the equation 9x+7y = 63 are x = 7, y = 0 and x = 0, y = 9.
      Put y = 0 in x - y = 10
                                x - 0 = 10
                               ∴ x = 10
      Put x = 0 in x- y = 10
                             0 - y = 10
                                  ∴ y = -10
      The solutions of the equation x - y = 10 are x = 10, y = 0 and x = 0, y = -10.
    ∴  There is no solution common to both the equations.        

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