# How to Solve Simultaneous Equations

## Solving Simultaneous Equations

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