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Year 9 Interactive Maths - Second Edition

Simultaneous Equations

Two linear equations in two variables are said to be simultaneous if they are considered at the same time.

Solution of Linear Simultaneous Equations

Simultaneous equations are solved exactly either by the substitution method or the elimination method.  An approximate solution can be found by using the graphical method.

Substitution Method

To solve simultaneous equations, find the value of  y in terms of x (or vice versa) for one of the two equations and then substitute this value into the other equation.

Example 1

Solve the following simultaneous equations by using the substitution method:


Label the equations as follows:

From (1) we have:

 y = 3x                       ...(3)

Substituting  y = 3x in (2) gives:

So, the solution is (2, 6).

The Graphical Method

The graphical solution of the simultaneous equations

is given by the point of intersection of the graphs.

Consider the graph of

It passes through the origin (0, 0) and the point (1, 3).

Consider the graph of x + y = 8.
intercept:  When y = 0, x = 8.
intercept:  When x = 0, y = 8.

The lines intersect at (2, 6).  So, the solution is (2, 6) as shown in the diagram.


Often the answer obtained with the graphical method is not exact.

Key Terms

simultaneous equations, substitution method, graphical method

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