Simultaneous Equations
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Simultaneous equations are two or more equations which are true for two or more unknowns. For example, x + y = 4 and x - 2y = 1 are simultaneous equations which are true for x = 3 and y = 1. When there are two unknowns, as there are here (x and y), then two equations are needed to find the unknowns. When there are 3 unknowns, 3 equations are needed, etc.

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?

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

Method 1: elimination
First form 2 equations. Let fish be f and chips be c.
We know that:
3f + 2c = 280    (1)
f + 4c = 260      (2)
Doubling (1) gives:
6f + 4c = 560 (3)
(3)-(2) is 5f = 300
\ f = 60
Therefore the price of fish is 60p

Substitute this value into (1):
3(60) + 2c = 280
\ 2c = 100
c = 50
Therefore the price of chips is 50p

Method 2: Substitution
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
\ 2x² - 7x + 6 = 0
\ (2x - 3)(x - 2) = 0
\ either 2x - 3 = 0 or x - 2 = 0
\ 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 below, y = 1 and x = 1.5 (approx.). These are the answers to the simultaneous equations.

Other Notes in this Category

1. Algebraic Long Division
2. Functions
3. Indicies
4. Logarithms
5. Partial Fractions
6. Reduction to Linear Form
7. Sequences
8. Series
9. Set Theory
10. Simultaneous Equations
11. Surds
12. The Binomial Series

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