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. Here are set of simultaneous equations. They are both linear equations:

Both equations are solved by the same values of x and y. x = 2 and y = 1 solve both equations.

Dictionary Definition

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

Where Does the Word Simultaneous Come From?

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

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Simultaneous Equations

Simultaneous equations are two or more equations that are solved by the same values of the variables. The most common types of simultaneous equations are two linear equations. In this mini-curriculum, you will learn about simultaneous equations.

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.
Here is an example of two simultaneous equations:


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Read more about simultaneous equations

Solving Simultaneous Equations

Solving simultaneous equations means finding the values of the variables ( x and y) that make both equations equal. There are different ways to solve simultaneous equations. In this mini-curriculum, you will learn how to solve simultaneous equations.

Solve Simultaneous Equations
To solve simultaneous equations, find the values of the variables that solve both equations.
Here is an example of a pair of simultaneous equations and the values of x and y that solve them:

2x + y = 4
x + 3y = 7



Solve Simultaneous Equations Using Elimination
To solve simultaneous equations using elimination, eliminate an unknown by adding or subtracting the equations.


Solve Simultaneous Equations Using Elimination Where Coefficients Differ
If the coefficients differ in the simultaneous equations (that is there are different numbers in front of the x or y), multiply one or both of the equations by a number to make one of the coefficients the same. Then add or subtract the equations.


Solve Simultaneous Equation Using Substitution
To solve simultaneous equations using substitution, subsitute an unknown from one equation into the other equation.
Here is an example of substituting y from one equation into another:


Solve Simultaneous Equation Using a Graph
To solve simultaneous equations using a graph, plot both equations on a graph and see where they cross.


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Read more about how to solve simultaneous equations
Read more about how to solve simultaneous equations using elimination
Read more about how to solve simultaneous equations using elimination (where the coefficients differ)
Read more about how to solve simultaneous equations using substitution
Read more about how to solve simultaneous equations using a graph