1. 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.

Explanation: N.B. Here we have found x in terms of y, instead of y in terms of x. Do which is the easier.

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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

:. x = 102/68 =51/34 =3/2 =11/2

substitute x = 11/2 in the equation 7x + 3y = 9
7(11/2) + 3y = 9

:. 101/2 + 3y = 9

:. 3y = 9 - 101/2

= - 11/2

:. y = -1/2

Check. when x = 11/2 and y = - 1/2

7x + 3y =7(11/2) + 3(-1/2) = 101/2 - 11/2 = 9.

11x - 5y = 11(11/2) - 5(-1/2)=161/2+21/2 = 19

Explanation: Not available

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3. 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)

:.  x = 12.

Substitute in (1) :         12 + y = 22

:. y = 10.

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

Explanation: Not available

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4. 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 ..

Explanation: Not available

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