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Simultaneous Equations (Mathematics Lesson)

What Are Simultaneous Equations?

A simultaneous equation is a set or system of equations containing multiple variables which has a solution that simultaneously satisfies all of the equations

For example, the set of simultaneous linear equations:



has a solution x = 1, y = 2.

Note: The letters - x and y - are called variables or unknowns.

The numbers in front of them are called co-efficients. Equation (1) is: (2 × x) + (1 × y) = 4.

Real Examples of Simultaneous Equations

An example of 2 simultaneous equations with 2 unknowns is:



The solution to these simultaneous equations is x = 1, y = 5.

An example of 3 simultaneous equations with 3 unknowns is:



The solution to these simultaneous equations is x = 2, y = 3, z = 4.

How to Solve Simultaneous Equations

To solve simultaneous equations, use:

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Algebra Lessons
Interactive Test
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Note
WHAT'S IN A NAME?

Simultaneous equations are so called because the solution solves all the equations at the same time, or simultaneously.

VARIABLES AND UNKNOWNS

The x and y in the simltaneous equations are called variables or unknowns.

They each have a value which solves all the simultanous equations.

The simultaneous equations must have the same variables as each other for them to be simultaneous equations. For example:



are not simultaneous equations.

NUMBER OF UNKNOWNS = NUMBER OF EQUATIONS

In the simultaneous equations shown to the left, there are 2 equations and 2 unknowns (x and y).

In general, to solve simultaneous equations, there must be as many independent equations as there are unknowns.

If there were 3 unknowns (e.g. x, y, and z), 3 equations would be needed to solve them.