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 MerriamWebster 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:
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.