Equations that must be solved at the same time are simultaneous equations.

**E.g.1**

The sum of two numbers is 6 and the difference is 2. Find the numbers.

Let the numbers be x and y.

x + y = 6 1

x - y = 2 2

1 + 2 => 2x = 8

x = 4

Sub in 1=> 4 + y = 6

y = 2

The numbers are x = 4 and y = 2.

**E.g.2**

The sum of two books and a pencil is £6.00. The difference of cost between 3 books and 2 pencils is £2.00. Find the cost of a book and a pencil.

Let the cost of a book be x and that of a pencil be y.

2x + y = 6 1

3x - 2y = 2 2

1 X 2 => 4x + 2y = 123

2 + 3 => 7x = 14

x = 2

Sub in 1

4 + y = 6

y = 2

The cost of a book and a pencil is £2.00 each.

**E.g.3**

If I double a number and add three times a second number, the answer is 1. If I multiply the first number by 3 and take away twice the second number, the answer is 8. Find the numbers.

Let the numbers be x and y.

2x + 3y = 11

3x - 2y = 82

From 1 - 3y => 2x = (1 - 3y)

x = (1 - 3y)/2

Sub in 2 => 3(1 - 3y)/2 - 2y = 8

X 2 => 3(1 - 3y) - 4y = 16

3 - 9y - 4y = 16

3 - 13y = 16

-3 => -13y = 13

y = -1

Sub in 1 => 2x - 3 = 1

+ 3 => 2x = 4

x = 2

The numbers are x = 2 and y = -1.

**E.g.4**

The sum of twice the cost of a box biscuits and the cost of a box chocolates is £8.00. The difference between the cost of box of chocolates and the box of biscuits is £1.00. Find the cost of each.

Let the cost of the box chocolates and the box of biscuits be y and x respectively.

2x + y = 8 1

y -x = 1 2

1 => y = 8 - 2x

Sub in 2 => y = 8 - 2x - x = 1

8 - 3x = 1

-8 => -3x = -9

:- -3 => x = 3

Sub in 1 => 6 + y = 8

-6 => y = 2

The cost of box of chocolates =£2.00 and that of biscuits = £3.00.

These equations are simultaneous as there are two unknowns in them; since one of the unknown is in quadratic form, they are quadratic too. Therefore, these equations have two sets of solutions, one for each unknown.

**E.g.1**

The equation of a circle is x^{2} + y^{2} = 45. It intersects with, y = 2x, at two points. Find the coordinates of the points of intersection.

x^{2} + y^{2} = 45 1

y = 2x2

Sub y in 1 => x^{2} + 4x^{2} = 45

5x^{2} = 45

:- 5 => x^{2} = 9

x = ± 3

Sub in 2 => y = ±6

Solutions: (3,6); (-3,-6)

**Now, in order to complement what you have just learnt, work out the following questions:**

- The average of two numbers is 11. The difference is 18. Find the numbers.
- A straight line passes through (3 , -4) and (5 , 8). Find the equation of the line.
- The cost of 3 DVD's and 4 CD's is £62.00. The cost of 4 DVD's and 3 CD's is £64.00. Find the cost of each.
- A diver swims downstream a distance of 40 miles in two hours. If he swims upstream, he can only move 16 miles during the same time. Find his swimming speed and the speed of the river.
- Half the difference of two numbers is 8. The average of the numbers is 12. Find the numbers.
- Nicole has £20 and £5 notes in her hand bag. The amount of money she has in the bag is £340.00. Find the number of notes of each type.
- There are two angles on a straight line. One angle is 15 more than twice the other. Find the size of each angle.
- The sum of ages of an uncle and his nephew two years ago was 40. In two years time from now, the age of the uncle will be three times that of his nephew by then. Find their ages in 7 years time.

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