**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.

solving simultaneous equations using elimination

## Substitution

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

solving simultaneous equations using substitution

## Graph

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

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**).

**3**unknowns (e.g.

**x**,

**y**and

**z**), we would need

**3**equations to solve them.

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