Simultaneous Equations
(KS4, Year 10)

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Simultaneous equations are a set of several equations with several unknowns. There are a set of values of these unknowns which solve all the equations at the same time.

Dictionary Definition

The Merriam-Webster dictionary defines simultaneous equation as "satisfied by the same values of the variables."

A Real Example of Simultaneous Equations

It is easier to understand simultaneous equations with an example. The most common type of simultaneous equations are a set of 2 linear equations, where the unknowns are the variables x and y: solve_simultaneous_equations

Both of these equations are linear. They show a straight line when plotted on a graph.
Both of these equations have two unknowns (or variables) : x and y. They can take different values.
Each equation has different coefficients and constants. In the top equation, the coefficient of x is 2. The coefficient of y is 1. The constant is 4. In the bottom equation, the coefficient of x is 1. The coefficient of y is 3. The constant is 7.

How to Solve Simultaneous Equations

The solution to simultaneous equations are the values of x and y which solve the equations simultaneously. The solution to the example above is x = 1 and x = 2: simultaneous_equations_solution

solving simultaneous equations

More Real Examples of Simulaneous Equations

Simultaneous equations do not have to be 2 linear equations.

3 (or More) Simultaneous Equations

It is possible for 3 or more linear equations to be solved simultaneously. 3 equations with 3 unknowns (x, y and z) are shown below. There are as many equations as there are unknowns (see Note.) simultaneous_equations_more_than_2

These simultaneous equations are solved by x = 1, y = 2 and z = 3.

Linear and Quadratic Equations

It is possible for one equation to be linear and another to be a quadratic equation. simultaneous_equations_linear_and_quadratic

Because of the quadratic equation, there will be 2 pairs of values of x and y that solve the equations simultaneously. These equations are solved by x = 0, y = 0 and x = 1, y = 1.

What's in a Name?

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

Number of Unknowns = Number of Equations

When there are 2 sets of linear equations:

there are 2 equations.
there are also 2 unknowns (x and y).

To solve simultaneous equations, there must be as many equations as there are unknowns. If there were 3 unknowns (e.g. x, y and z), we would need 3 equations to solve them.