Question:Solve the equations
                    
                      `3x + 4y = 5`.......................................... (i)
                      `2x + 6y = 5` ...........................................(ii) 

From (i),                      3x = 5 - 4y

                                    x = `(5 - 4y)/ 3`.

From (ii),                      2x = 5 - 6y

                                     x = `(5 - 6y)/2` .

                                   `:. (5 - 4y)/3` = `(5 - 6y)/2` .

Multiply throughout by 6 :
                                10 - 8y = 15 - 18y.
                              `:.` 18y - 8y = 15 - 10.
                              `:.` 10y = 5 and y = `1/2`.

Substitute this value of y in x = `(5 - 6y)/2`.

                                        x = `(5 - 3)/2`= `2/2`= 1.

The solution is x = 1; y =`1/2`.

Question:Solve the equations 
          7x + 3y = 9. . . .  . . . . . . . . . . . . . .  .(1)
          11x - 5y =19. . . . . . . . . . . . . . . . . . .(2) 

Eliminate y from the questions.

Multiply (1) by 5:    35x + 15y=45
Multiply (2) by 3:    33x - 15y=57

add:                     68x        =102

                        `:.` x = `102/68` =`51/34` =`3/2` =1`1/2`

substitute x = 1`1/2` in the equation 7x + 3y = 9
                                         7(1`1/2`) + 3y = 9

                                     `:.` 10`1/2` + 3y = 9 

                                                    `:.` 3y = 9 - 10`1/2`

                                                               = - 1`1/2`
                                                 
                                                           `:.` y = -`1/2`


Check. when x = 1`1/2` and y = - `1/2`

 7x + 3y =7(1`1/2`) + 3(-`1/2`) = 10`1/2` - 1`1/2` = 9.

11x - 5y = 11(1`1/2`) - 5(-`1/2`)=16`1/2`+2`1/2` = 19

Question:The perimeter of a rectangle is 44 m . The length is 2 m greater than the width. Find the length and width  of the rectangle.. 

The perimeter of the rectangle is x 2(x+y) meters.
                                                 
                                                  `:.` 2(x+y) =44.

Divide both sides of this equations by 2;
                                            
                                              x+y = 22. . . . . . . . . . . . . . . . . . . . .(1) 


We are also told that the length is 2 m greater than the width.

                                           `:.` x = y+2, 
                                                x - y = 2 . . . . . . . . . . .  . . . . . . . . .(2)

we have the two equations 
                             
                                      x + y = 22 . . . . . . . . . . . . .(1)
                                      x - y  = 2  . . . . . . . . . . . . . (2)

add:                                   2x = 24.
                 
                              `:.`  x = 12.

Substitute in (1) :         12 + y = 22
                
                             `:.` y = 10.

The length of the rectangle is 12 m ; its width is 10  m.

Question:The cost of 5 oranges and 4 lemons  is 23 new pence.The  cost of 3 oranges and 5 is 19 new pence.find the cost of an orange.. 

The cost of 5 oranges and 4 lemons is (5x + 4y) new pence.
The cost of 3 oranges and 5 lemons is (3x + 5y) new pence.

                                          `:.  5x + 4y =23  . . . . . . . . . . . . . . . . . . .(1) 
                                                  3x + 5y =19  . . . . . . . . . . . . . . . . . . .(2)`

Multiply (1) by 3 :        15x + 12y = 69
Multiply (2)  by 5 :       15x + 25y = 95

Subtract :                     - 13y = - 26  
                                  `:.` y = 2 

substitute in (1)            5x + 8 = 23

                                `:.`5x =23 - 8

                                           = 15

                                  `:.` x = 3.

The cost of an orange is 3 new pence ; the cost of a lemon is  2 new pence ..