# 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