 # Simultaneous Equation- Meaning, Methods and Examples

This article will give you a detailed illustration of simultaneous equation including the meaning, methods, examples and detailed steps for solving it. At the end of this article you will be able to solve simple and complex exercises involving simultaneous equation.

## What Is Simultaneous Equation?

Simultaneous equation is an equation wherein two or more equations in two or more unknown variables are to be solved at the same time and the values of the unknowns satisfy the equations.

Simultaneous equation can also be called simultaneous linear equations when all the equations are linear, that is, the power or index of the unknowns is 1. For example

ay + bx = c……………..(1)

dx + ey = f……………..(2)

Where: a,b,c,d,e and f are constants and x and y are unknown variables.

## What Are The Methods Of Solving Simultaneous Equation?

There are three (3) basic methods used in solving simultaneous equations. All three methods give the same answer and there are very easy for you to assimilate. We are going to list and explain each of them and also solving various academic exercises on them.

The methods include:

• Substitution method
• Elimination method
• Graphical method

However, this article will only concentrate on the substitution and elimination methods.

### Substitution Method:

This is the most widely used method for solving basic simultaneous equation exercises. It involves reducing the two given equation in two unknown variables to one equation in one variable by substituting one variable in terms of the other and then solving the resulting linear equation to get the value of the unknown variable.

This value is then substituted in the other equation to get the value of the other unknown variable. This method is far easier for you to use when the coefficient(s) of any of the unknown is unity that is 1.

#### Example 1

Solve the simultaneous equation below using substitution method

3x + y = 1 and 2x – 3y = 8.

#### Solution

##### Step 1: put the question in double equation form

3x + y = 1………………..(1)

2x – 3y = 8………………(2)

Step 2: check if any of the unknown has a coefficient of 1. It could be used easily.

In the above equations, y in equation on has a coefficient of 1 and therefore it is selected. (N/B: if none of the equations has a coefficient of 1, you can select any of the unknown and proceed)

Step 3: make y the subject of the fomular, that is, express y in terms of x and call it eqn 3.

3x + y = 1

y = 1 – 3x…………..(3)

step 4: substitute the value of y (1 – 3x) in eqn 2.

2x – 3y = 8………………(2)

2x – 3(1 – 3x) = 8

Open brackets

2x – 3 + 9x = 8

Collect like terms

2x + 9x = 8 + 3

11x = 11

Divide both sides by the coefficient of x

11x/11 = 11/11

x = 1

##### Step 5: Substitute 1 for x in eqn (3)

y = 1 – 3x…………..(3)

y = 1 – 3(1)

Then, y = 1 – 3 = -2

Therefore, x = 1 and y = -2

To confirm if the answer to the simultaneous equation is correct replace the unknown with their known values and check if it will give you the answer.

3(1) + (-2)

3-2 = 1

2(1) – 3(-2)

2+ 6 = 8

Example 2

Solve the simultaneous equation below:

2a + 3b = 6 and 3a + 4b = 8

#### Solution

Just like you did in the first exercise, you will follow all the steps sequentially.

2a + 3b = 6 ……………………(1)

3a + 4b = 8 ……………………(2)

From eqn 1, you will make a the subject of the formular

2a = 6 – 3b

a = (6-3b)/2………………………….(3)

substitute the value of a in eqn 2

3a + 4b = 8

3(6-3b)/2 + 4b = 8

(18-9b)/2 + 4b = 8

9 – (9b)/2+ 4b = 8

Collect like terms

4b – (9b)/2 = 8 -9

4b – (9b)/2 = -1

Multiply both sides by 2

8b -9b = -2

-b = -2

b = 2

Substitute the value of b in eqn 3

a =(6-3b)/2………………………….(3)

a =(6-3(2))/2

a = (6-6)/2

a = 0

Therefore, a = 0 and b = 2

2a + 3b = 6 ……………………(1)

2(0) + 3(2)

3a + 4b = 8 ……………………(2)

3(0) + 4 (2)

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### Elimination Method:

In this method, you have to get rid of or eliminate one of the unknown variable in order to get the other variable. You can easily do this by making the coefficient of any of the variable unity, that is, the same so that they can cancel out easily.

In the above example 2, elimination method is easier to apply than substitution method. As a note, anytime the coefficient of any of the unknown variable in any given equation is 1, use substitution method. Whenever the coefficients of the unknown variables are more than 1, use the elimination method.

We are going to apply the elimination method of solving simultaneous equations in the following exercises.

#### Example 1

Solve the simultaneous equation 7x + 4y = 27 and 4x + 3y = 14 using the elimination method.

#### Solution

Step 1: Write out the question in two equation forms

7x + 4y = 27……………………eqn 1

4x + 3y = 14 ………………….eqn 2

Step 2: Eliminate y in the two equations by multiplying eqn 1 by the coefficient of y in eqn 2 and doing the same thing for eqn 2.

Coefficient of y in eqn 1 = 4

Coefficient of y in eqn 2 = 3

3 (7x + 4y = 27)……………………eqn 1

4 (4x + 3y = 14) ………………….eqn 2

Step 3: Open brackets and proceed to simplify

21x + 12y = 81……………..eqn 3

16x + 12y = 56……………eqn 4

Step 4: Substract eqn 4 from eqn 3

5x = 25

x = 5

To get the value of y, eliminate x in eqn 1 and 2 by following the same steps

4 (7x + 4y = 27)……………………eqn 1

7(4x + 3y = 14) ………………….eqn 2

28x + 16y = 108……………..eqn 5

28x + 21y = 98………………eqn 6

Substract eqn 6 from eqn 5

-5y = 10

y = -2

Therefore, x = 5 and y equals to -2

We will take another example to further enhance our understanding of solving simultaneous equation using the elimination method.

#### Example 2

Solve the simultaneous equation below using elimination method.

2x + 2y = 14 and 5x – 3y = 5

#### Solution

2x + 2y = 14…………. Eqn 1

5x – 3y = 5………….eqn 2

###### Eliminating x

5 (2x + 2y = 14)…………eqn 3

2 (5x – 3y = 5)…………eqn 4

10x + 10y = 70

10x – 6y = 10

4y = 60

y = 15

###### To solve for y

-3 (2x + 2y = 14)…………eqn 5

2 (5x – 3y = 5)…………eqn 6

-6x – 6y = -42

10x -6y = 10

Simplify

-16x = -52

x = 3.25

##### NOTE: You must be very mindful of your arithematic signs when you are simplifying equations. Also apply the rule of BODMAS when simplifying simultaneous equation.

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