Sunday, March 31, 2013

Solve the following pair of linear equations by the substitution method.


EXERCISE 3.3
1. Solve the following pair of linear equations by the substitution method.
(i) x + y = 14  ; x y = 4                                (ii) s t = 3 ; s/3 + t/2 = 6
(iii) 3x y = 3 ; 9x – 3y = 9                          (iv) 0.2x + 0.3y = 1.3  ; 0.4x + 0.5y = 2.3
(v)√2 x+  √3y =  0 ; √3 x -  √8y =  0 (vi)3/2 x -  5/3y = -2 ; x/3 +  y/2 = 13/6
(i)              x + y = 14  ; x y = 4                      
solve first equation
x + y   = 14
x         = 14 - y  …………..(1)
plug this value in equation second we get
x – y   = 4
14 – y   - y     = 4
Add 14 both side we get
- 2 y = 4 - 14
- 2 y = - 10
     = -10/-2
y      = 5
Plug y = 5 in equation first we get
      = 14 – y
      = 14 – 5
      = 9
Answer x  = 9 , y = 5
(ii)            s t = 3 ; s/3 + t/2 = 6
                       
 Solve first equation
s t     = 3 
s          = 3+t …………(1)
Plug this value in equation second we get
s/3 + t/2          = 6
 
Multiply by 6 to remove all denominator we get
6 + 2t  +  3t     =  36
5t         = 36 – 6
5t         = 30
                                                Divide by 2 we get
                                    t           = 30/5
                                    t           =6
Plug the value in equation first we get
                                    s          = 3 +6
                                    s          = 9
                        Answer s  = 9 , t = 6
(iii)           3x y = 3 ; 9x – 3y = 9      
solve first equation
3x y = 3
3x        =3 +y
 x         = (3 + y)/3  …………..(1)
Plug this value in equation second we get

9 is always equal to 9
So it is a true condition and in such condition
Equations will have many or infinity solutions
(iv)           0.2x + 0.3y = 1.3      …..(1)  ;
0.4x + 0.5y = 2.3                               ...(2)
0.2x + 0.3y = 1.3 
Solve first equation we get
0.2x  = 1.3 – 0.3y
Divide by 0.2 we get
x        = 1.3/0.2   - 0.3/0.2
x         = 6.5 – 1.5 y              …(3)
Plug the value in equation second we get
0.4x + 0.5y    = 2.3
(6.5 – 1.5y)*0.4x + 0.5y        =  2.3
2.6 – 0.6y + 0.5y       = 2.3
-0.1 y  = 2.3 – 2.6
 y        = - 0.3/-0.1
 y        = 3
Plug this value in equation 3 we get 
                                                x         = 6.5 – 1.5 y
                                                x         = 6.5 – 1.5(3)
                                                x         =6.5  - 4.5
                                                x         =2
                                                                                                Answer x = 2, y = 3
                                                                       
(v)            √2 x+  √3y =  0                                             ...(1)
              √3 x -  √8y  = 0                                             ... (2)  
√2 x+  √3y =  0
Subtract √3y both side we get
√2 x     =  -√3y
Divide by √2
       ……(3)
Plug the value of x in equation first we get
√3 x -  √8y =  0
                                       
   
Plug this value in equation (3) we get 
Our answer   x = 0 and y = 0
(vi)           3/2 x -  5/3y = -2                                                       ……(1)
 x/3 +  y/2 = 13/6                                                       .......(2)
Solve first equation
           3/2 x -  5/3= -2
Multiply by 6 to remove all denominator we get
9 x – 10 y       = -12
9x      = -12 + 10y
Divide by 9 we get
x          = (-12 + 10 y)/9                      …..(3)
plug this value in equation (2) we get

 
Multiply by LCM of denominator  = 27* 2  = 54 we get
2(-12 +10 y ) +  27 y= 117
-24 +  20 y + 27 y      = 117
47 y     = 117  +  24
Y        = 141/47         = 3
Plug this value in equation (3) we get
x          = (-12 + 10*3)/9
                                x         =18/9
x          = 2
Answer x = 2 and y = 3

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