Simultaneous Equations

Definition of Simultaneous Equations

 A pair of "Simultaneous equations" is two equations which are both true at the same time. You have two equations which have two unknowns to be found.

Example

 A man buys 3 fish and 2 chips for £2.80

A woman buys 1 fish and 4 chips for £2.60

How much are the fish and how much are the chips?

 First we form the equations. Let fish be f and chips be c.

We know that:

3f + 2c = 280    (1)

f + 4c = 260      (2)

 There are two methods of solving simultaneous equations. Use the method which you prefer:

Elimination

 This involves changing the two equations so that one can be added/ subtracted from the other to leave us with an equation with only one unknown (which we can solve). We can 'change' the equations by multiplying them through by a constant- as long as we multiply both sides of the equation by the same number it will remain true.

 In our above example:

 Doubling (1) gives:

6f + 4c = 560 (3)

 Since equation (2) has a 4c in it, we can subtract this from the new equation (3) and the c's will all have disappeared:

(3)-(2) gives 5f = 300  ∴ f = 60

Therefore the price of fish is 60p

 So we can put f=60 in either of our original equations. Substitute this value into (1): 3(60) + 2c = 280 ∴ 2c = 100

c = 50

Therefore the price of chips is 50p

Substitution

 The method of substitution involves transforming one equation into x = (something) or y = (something) and then substituting this something into the other equation.

 So,Rearrange one of the original equations to isolate a variable.

Rearranging (2): f = 260 - 4c

Substitute this into the other equation:3(260 - 4c) + 2c = 280∴ 780 - 12c + 2c = 280 ∴ 10c = 500

∴ c = 50

Substitute this into one of the original equations to get f = 60 .

 Harder simultaneous equations

 To solve a pair of equations, one of which contains x2, y2 or xy, we need to use the method of substitution.

Example

 2xy + y = 10  (1)

x + y = 4        (2)

Take the simpler equation and get y = .... or x = ....

from (2), y = 4 - x    (3)

this can be substituted in the first equation. Since y = 4 - x, where there is a y in the first equation, it can be replaced by 4 - x .

sub (3) in (1), 2x(4 - x) + (4 - x) = 10

∴ 8x - 2x2 + 4 - x - 10 = 0

∴ 7x - 2x2 - 6 = 0

∴ 2x2 - 7x + 6 = 0  (taking everything to the other side of the equals sign)

∴ (2x - 3)(x - 2) = 0

∴ either 2x - 3 = 0 or x - 2 = 0

therefore x = 1.5 or 2 .

 Substitute these x values into one of the original equations.

When x = 1.5,  y = 2.5

when x = 2, y = 2

Using Graphs

 You can solve simultaneous equations by drawing graphs of the two equations you wish to solve. The x and y values of where the graphs intersect are the solutions to the equations.

Example

 Solve the simultaneous equations 3y = -2x + 6 and y = 2x -2 by graphical methods.

  

  From the graph, y = 1 and x = 1.5 (approx.). These are the answers to the simultaneous equations.