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

y      = -10/-2

y      = 5

Plug y = 5 in equation first we get

x       = 14 – y

x       = 14 – 5

x       = 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 -  5/3y  = -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