The Lesson

Simultaneous equations are a set of several equations with several unknowns. In this lessons, we will be looking at a set of 2 linear equations, where the unknowns are the variables x and y:

There are a set of values of these unknowns which solve all the equations at the same time. The solution to the simultaneous equations shown above are:

Understanding Solving Simultaneous Equations

It is easier to understand solving simultaneous equations with an example.
2x + y = 4 ... (1) x + 3y = 7 ... (2)
x and y are variables. They can take different values. Let's try different pairs of values of x and y.

x = 2, y = 0

Substitute x = 2 and y = 0 into both equations and see if the left hand side of the equation equals the right hand side of the equation:
Equation (1) ... 2(2 ) + ( 0 ) = 4 Equation (1) ... 2 × 2 + 0 = 4 Equation (1) ... 4 = 4 ? ✔
Equation (2) ... 2 + 3( 0 ) = 7 Equation (1) ... 2 + 3 × 0 = 7 Equation (1) ... 2 = 7 ? ✖
x = 2 and y = 0 solves Equation (1) but does not solve Equation (2). It does not solve both equations at the same time. It is not a solution to the simultaneous equations.

x = 4, y = 1

Substitute x = 4 and y = 1 into both equations and see if the left hand side of the equation equals the right hand side of the equation:
Equation (1) ... 2(4 ) + ( 1 ) = 3 Equation (1) ... 2 × 4 + 1 = 3 Equation (1) ... 9 = 3 ? ✖
Equation (2) ... 4 + 3( 1 ) = 7 Equation (1) ... 4 + 3 × 1 = 7 Equation (1) ... 7 = 7 ? ✔
x = 4 and y = 1 solves Equation (1) but does not solve Equation (2). It does not solve both equations at the same time. It is not a solution to the simultaneous equations.

x = 1, y = 2

Substitute x = 1 and y = 2 into both equations and see if the left hand side of the equation equals the right hand side of the equation:
Equation (1) ... 2(1 ) + ( 2 ) = 4 Equation (1) ... 2 × 1 + 2 = 4 Equation (1) ... 4 = 4 ? ✔
Equation (2) ... 1 + 3( 2 ) = 7 Equation (1) ... 1 + 3 × 2 = 7 Equation (1) ... 7 = 7 ? ✔
x = 1 and y = 2 solves Equation (1) and Equation (2). It solves both equations at the same time. It is a solution to the simultaneous equations. x = 1 and y = 2 solve the simultaneous equations.

How to Solve Simultaneous Equations

There are 3 ways to solve simultaneous equations.

Elimination

Simultaneous equation can be solved by eliminating one of the unknowns by adding or subtracting linear combinations of the equations.


Read more about solving simultaneous equations using elimination

Substitution

Simultaneous equation can be solved by substituting one of the equations into the other.


Read more about solving simultaneous equations using substitution

Graph

Simultaneous equation can be solved by plotting each equation on a graph and seeing where they meet.


Read more about solving simultaneous equations using a graph

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

In the simultaneous equations in this lesson:
  • there have been 2 equations.
  • there have also been 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.