# Equations » Simultaneous equations

## Contents

What is a set of simultaneous equations?
Method 1: Rearranging
Method 2: Substitution

## What is a set of simultaneous equations?

A set of simultaneous equations is also known as a system of equations

A set of simultaneous equations looks like this: 2x + 4y = 100
5x – 2y = 64

Normally it is not possible to solve an equations with more than one variable. However, in a set of simultaneous equations you have two or more equations about the same subject. Because they are about the same subject, it is possible to calculate the solutions.

Example
There are 750 pupils attending a certain school. If you multiply the number of boys times two and subtract 120, you have the number of girls. How many boys and how many girls attend this school? b + g = 750
2b – 120 = g

There are three different ways of solving a set of simultaneous equations. Below every method is explained using an example.

## Method 1: Rearranging

Solve the following set of simultaneous equations. 2x + 4y = 100
5x – 2y = 64

Step 1: Rearrange both equations to a formula.

 2x + 4y = 100 4y = –2x + 100 y = –12x + 25
 5x – 2y = 64 –2y = –5x + 64 y = 212x – 32

Step 2: Equate both formulas.

 –12x + 25 = 212x – 32 –3x = –57 x = 19

Step 3: Calculate the other variable.
y = –12x + 25 = –12 × 19 + 25 = 1512

## Method 2: Substitution

Solve the following set of simultaneous equations. 4m – n = 80
2m – 5n = 220

Step 1: Rearrange one of the equations to a formula.
Rearranging the first equation to n = is the fastest.

 4m – n = 80 –n = –4m + 80 n = 4m – 80

Step 2: Substitute this formula in the other equation.

 2m – 5n = 220 2m – 5(4m – 80) = 220 2m – 20m + 400 = 220 –18m = –180 m = 10

Step 3: Calculate the other variable.
n = 4m – 80 = 4 × 10 – 80 = –40

Solve the following set of simultaneous equations. –7x + 2y = –60
10x – 4y = 136

Step 1: Rearrange one of the equations so you get a term that is the opposite of the like term in the other equation. In other words: If one equation has +2y, make sure the other has –2y.

 10x – 4y = 136 5x – 2y = 68

Step 2: Add the two equations. –7x + 2y = –60 5x – 2y = 68  + –2x + 0y = 8 –2x = 8 x = –4

Step 3: Calculate the other variable. You have to use one of the equations for this.

 –7x + 2y = –60 –7 × –4 + 2y = –60 28 + 2y = –60 2y = –88 y = –44

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