# Simultaneous Equations

Updated: Jun 16, 2021

Math Lobby will be going through Simultaneous Equations today. Simultaneous equations are tested yearly in the O Level E Math, O Level A Math, N Level Math and N Level A Math examination. It will be covered in Secondary 2 and 3 Math** **syllabus.

Without further ado, let us learn how to solve simultaneous equations today!

__In this note, you will learn:__

**1) What are “simultaneous equations”?**

**2) What are the types of method to solve simultaneous equations?**

**(Substitution and Elimination)**

__What are “simultaneous equations”?__

__What are “simultaneous equations”?__

By definition, simultaneous equations are equations that contain **two or more unknown variables** that have the same values in each equation.

Given an example of an algebraic equation,

**x + 2y = 6**

Usually when we have an algebraic equation with only one variable (let’s say **3x + 5 = 20**), we know that we have to isolate **x** (also known as **“making x the subject”**) and by shifting 5 over to the right side, and divide the whole equation by 3 to solve for x.

3x + 5 = 20

3x = 20 – 5

3x = 15

Therefore, x = 5

However, when we have **2 unknown variables**, it is impossible to do this! So, what should be done to know both the possible values of **x** and **y**?

__What are the types of method to solve simultaneous equations?__ (Elimination and Substitution)

__What are the types of method to solve simultaneous equations?__(Elimination and Substitution)

Fortunately, there are ** two** ways of approach for solving simultaneous equations. The first method is called the

**elimination**method. The reason why it is called the elimination method is because what we are essentially doing is making the constant of a chosen variable the same for both the equations. Let’s see how it works out:

*Method 1: Elimination Method*

*Method 1: Elimination Method*Given an example of two simultaneous equations,

## x + 2y = 6 ---- ①

# 4x + 10y - 8 = 20 ---- ②

First of all, we have to choose which variable to **eliminate**, then make its constant for both equation the same. The general rule of thumb is to chose the variable that is the easiest to deal with, which in this case is the x-variable in equation ①. Now, we will have to make its constant the same as the constant of the x-variable in equation ②.

Since the constant of the x-variable in equation ② is 4, and the constant of the x-variable in equation ① is 1, we must hence multiply equation ① by 4 in order for it to give us the same number of x in equation ②.

Let’s try it!

If we multiply ① by 4, we will get:

### (x + 2y) x 4 = (6) x 4

We will then name this equation, ③:

## 4x + 8y = 24 ---- ③

Then, we can eliminate the x-variable by using equation ② to __SUBTRACT__ equation ③ (hence, the name “elimination”).

How we do this is by subtracting the left side of equation ③ with the left side of equation ②, and the right side of equation ③ with the right side of equation ②. Let’s see how it’s done:

### (4x + 10y – 8) – (4x + 8y ) = 20 – 24

Left/right side of equation ② subtract left/right side of equation ③

And therefore, we will end up with:

### 2y - 8 = -4

### 2y = 4

### And hence, y = 2

To solve for the value of the x-variable, just simply substitute y = 2 back into any of the equations.

For example, if we use equation ①, x + 2y = 6

### x + 2(2) = 6

### x + 4 = 6

### Therefore, x = 6 – 4 = 2

If you are unsure of whether if the answer you got is the correct values of x and y, just simply substitute both values back into the equation and see if it is mathematically logical!

__Math Lobby Advice: How to double check if your answers are correct!__

Let’s double check our answers by substituting both x = 2 and y = 2 into equation ②,

### 4x + 10y – 8 = 20

### 4(2) + 10(2) – 8 = 20

### 8 + 20 – 8 = 20

### Therefore, 20 = 20

And there you have it! We just double checked our answers and it is indeed correct! Just remember: the best way to ace your mathematics test is to practice, practice and practice!

*Method 2: Substitution Method*

*Method 2: Substitution Method*Next, we will be talking about the second method which is the **substitution**** method**. For the substitution method, it is a much simpler method to get the hang of, and once you are comfortable with it, solving simultaneous equations will be a piece of cake!

In order to begin with the substitution method, we must first make a chosen variable the subject. Then, use that equation and substitute it into the other equation, (hence, the name “substitution”) which will leave us with only one variable to solve!

Let’s try the same example with this method:

## x + 2y = 6 ---- ①

## 4x + 10y - 8 = 20 ---- ②

First, we need to start off by making a chosen variable the subject. Now if you look closely, we can see that in equation ①, we have a singular x-variable. It seems the easiest to deal with, so let’s make x the subject and name the new equation, ③.

### x + 2y = 6

## x = 6 – 2y ---- ③

Then, we will substitute this new equation ③ into the other equation, which is equation ②.

### Sub ③ into ②, 4(6-2y) + 10y – 8 = 20

### 24 – 8y + 10y – 8 = 20

### 2y + 16 = 20

### 2y = 20 – 16

### And therefore, y = 2

Now, to solve for x, it is exactly the same! Just substitute y = 2 into equation ① and you will the value of x! Isn’t it easy?

Since you have learnt how to solve simultaneous equations using both the **elimination method** and the **substitution method**, why not put it to the test?

Just remember: the __best way to ace your mathematics test__** **is to practice, practice and practice!

There are a few questions below for you to try:

__Simultaneous Equations Questions__

Given the simultaneous equations, solve for the values of x and y.

## 1. 6x + 3y = 18

## 4y + 9x + 3 = 55

## 2. 18y + 5x = 90

## 15x + 18y – 162 = -72

**Ans: 1) x = 28, y = -50 2) x = 0, y = 5**

Let’s take this up a notch!

__Simultaneous Equations Questions (Advanced) __

Given the simultaneous equations, solve for the values of x and y.

## 16x + 4y = 8

## 9y2 + 4xy + 392x = -92

*Hint: You will end up with a quadratic equation to solve!

*Ans: x = -1, y = 6*

And that’s all for today, and I hope that everyone has a clear understanding of what simultaneous equations are, and how we can solve them with one of the two methods: **elimination** and **substitution**.

Remember to practice hard, which is applicable for every students regardless of stream:

O, N Level Elementary Mathematics, or O, N Level Additional Mathematics.

Never be afraid to ask for a ** mathematics tutor**’s help if you need to, because Math Lobby’s mathematics tuition center will always be here for you! Work hard, stay motivated and I wish all students a successful and enjoyable journey with Math Lobby!

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