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Revision:Simultaneous Equations

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Simultaneous Equations

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.


Contents

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

NB. elimination can only be done when each unknown has the same power in each equation. First form 2 equations. Let the cost of fish be f and the cost of chips be c.

We know that:

\displaystyle 3f + 2c = 280 \quad \quad \quad \quad (1)

\displaystyle f + 4c = 260 \quad \quad \quad \quad (2)


Doubling (1) gives:

\displaystyle 6f + 4c = 560 \quad \quad \quad \quad (3)


Then, \displaystyle (3)-(2) \Rightarrow 5f = 300

\displaystyle \Rightarrow f = 60.


Therefore the price of fish is 60p


Substitute this value into (1):

\displaystyle 3(60) + 2c = 280

\displaystyle 2c = 100

\displaystyle c = 50


Therefore the price of chips is 50p

Method 2: Substitution

Rearrange one of the original equations to isolate a variable.

Rearranging (2): \displaystyle f = 260 - 4c


Substitute this into the other equation:

\displaystyle 3(260 - 4c) + 2c = 280

\displaystyle 780 - 12c + 2c = 280

\displaystyle 10c = 500

\displaystyle 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 x^2, y^2 or xy, we need to use the method of substitution.


Example

\displaystyle2xy + y = 10 \quad \quad \quad \quad (1)

\displaystyle x + y = 4 \quad \quad \quad \quad (2)


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

from (2), \displaystyle y = 4 - x \quad \quad \quad \quad (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):

\displaystyle 2x(4 - x) + (4 - x) = 10

\displaystyle 8x - 2x^2 + 4 - x - 10 = 0

\displaystyle 2x^2 - 7x + 6 = 0

\displaystyle (2x - 3)(x - 2) = 0

either \displaystyle 2x - 3 = 0 or \displaystyle 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 below, y = 1 and x = 1.5 (approx.). These are the answers to the simultaneous equations.


(diagram of graph missing)



Comments

Missing graphs for graphical methods example.

This article covers the topic at GCSE level however it can also serve as useful revision for those studying the subject at A level.

Solving simultaneous equations by using matrices is missing.

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