What Are Simultaneous Equations?

What Are Simultaneous Equations?

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:

  • 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:

Read more about 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.)

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.

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.

See Also

What is an equation? What is a linear equation? What is a variable? What is a coefficient? What is a constant? Solving simultaneous equations What is a quadratic equation?