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© Matthew Pinkney 2003
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 (these satisfy both equations). 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- in fact in order to solve simultaneous equations, in general there must be as many different equations as there are unknowns.
A man buys 3 fish and 2 chips for £2.80
First form 2 equations. Let fish be f and chips be c.
Method 1: Elimination
In this method we add or subtract the equations to/from one another.
Doubling (1) gives:
Method 2: Substitution
Rearrange one of the original equations to isolate a variable.
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.
2xy + y = 10 (1)
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.
Solve the simultaneous equations 3y = -2x + 6 and y = 2x -2
by graphical methods.