# Simultaneous Equation- Meaning, Methods and Examples

This article will provide a detailed illustration of a simultaneous equation, including its meaning, methods, examples, and detailed steps for solving it. At the end of this article, you can solve simple and complex exercises involving simultaneous equations.

## What Is a Simultaneous Equation?

A simultaneous equation is an equation in which two or more equations involving two or more unknown variables are to be solved simultaneously, 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 Equations?

There are three (3) basic methods used in solving simultaneous equations. All three methods give the same answer and are very easy for you to assimilate. We will list and explain each of them and solve 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 equations 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 to use when the coefficient(s) of any of the unknowns is unity, that is, 1.

#### Example 1

Solve the simultaneous equation below using the 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 unknowns has a coefficient of 1. It could be used easily.

In the above equations, y in the 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 formula, that is, express y in terms of x and call it eqn 3.

3x + y = 1

y = 1 – 3x…………..(3)

Step 4: Substituting 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 unknowns with their known values and check if it gives you the answer.

3(1) + (-2)

3-2 = 1

2(1) – 3(-2)

2+ 6 = 8

Therefore, the answers are correct.

Example 2

Solve the simultaneous equation below:

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

#### Solution

As 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 formula

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

To check if your answer is correct

2a + 3b = 6 ……………………(1)

2(0) + 3(2)

3a + 4b = 8 ……………………(2)

3(0) + 4 (2)

Therefore, your answers to the unknowns were correct.

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### Elimination Method:

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

In the above example 2, the elimination method is more accessible than the substitution method. As a note, whenever the coefficient of any unknown variable in any given equation is 1, use the substitution method. Use the elimination method whenever the coefficients of the unknown variables are more than 1.

The following exercises will apply the elimination method of solving simultaneous equations.

#### 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: Subtract 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

Subtract eqn 6 from eqn 5

-5y = 10

y = -2

Therefore, x = 5 and y equals to -2

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

#### Example 2

Solve the simultaneous equation below using the 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 mindful of your arithmetic signs when simplifying equations. Also, the rule of BODMAS should be applied when simplifying simultaneous equations.

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