# Solving Simultaneous Equations(KS4, Year 10)

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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.
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).
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.

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