Understanding Equations and Formulas in Maths

What is an Equation?

Equations are vital parts of our lives. We use equations not only in subjects like Mathematics and Science but also to calculate the price, debt, tax, interest, etc. The best example of an equation is “ 5 + 5 = 10”. But exactly, what is an equation?

An equation is an analytical statement that states that two things are equal. An equation includes either term or expression. In Mathematics, an equation is defined as equality including one or more variables. Solving equations means determining which values of the variables make the equation true. In this case, variables are considered as unknown and the values which satisfy the equality are known as solutions. An equation differs from identity in that equation is not certainly valid for all possible values of the variables.

The (“=”) symbol which can be seen in every equation, was introduced by Robert Recorde in 1557, who stated that nothing can be more equal than the parallel straight lines with similar length.

Define Equation

An equation is a Mathematical statement with an ‘equal to’ sign between two algebraic expressions that have similar values.

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For example, 5x + 9 is the expression on the left-hand side which equals to the expression 24 on the right-hand side.

Look at the following examples of equations. This will give you a clear idea of what the equation is in Maths.

Equation Examples

What is Formula?

In Mathematics, a formula is a fact or rule expressed with a Mathematical symbol.

The formula generally includes:

• An ‘ equal to’ symbol.

• Two or more variables (x, y, etc)

For Example,

The perimeter of a rectangle formula is

Perimeter = 2 (Length + Breadth)

if the length and width of the rectangle are ‘a’ and ‘b’ respectively, the formula of its perimeter is

Perimeter = 2 ( a + b)

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Formula Without “ Equal to” Symbol

Sometimes, the formula can be expressed without ‘ equal to” sign

For Example,

But this can also be expressed with an ‘ equal to’ symbol as it can also be written as Perimeter of a square = 4 x side.

What is the Subject of a Formula?

The subject of a formula is the single variable which is expressed in terms of other variables included in the formula.

Formulas are written so that a single variable that is the subject of the formula is written on the left-hand side of the equation and everything else goes on the right side of the equation.

For Example,

In the formula v = u + at, v is considered as the subject of the formula.

How to Change the Subject of a Formula?

To change the subject of a formula, items in the formula need to be rearranged, so that new variables become the subject of the formula. Knowledge of solving equations and inverse operations is important to have while changing the subject of a formula.

For example, 

In formula A= bh, area (A) is the subject of the formula which means it is the area that has to be calculated.

If the area and height of a rectangle are given and you are asked to calculate the base of a rectangle, the formula A= bh will not be helpful to calculate the area of the rectangle as now ‘b’ has to be calculated.

To calculate b that is the base of a rectangle, the formula A = bh has to be reordered to make bas the subject of the formula.

To make bas the subject of the formula, b needs to be isolated. In the above formula, the variable b is multiplied by the variable h. The inverse of multiplying the variable by h , is dividing it by h.

In the formula A = bh   we will divide both the side by h to isolatebas shown below:

A/h = bh/h

A/h = b

The variable b is now the subject of a formula.

Now, we will use the formula A/h = b, to calculate the base of a rectangle. 

Solved Examples

1. Rearrange the volume of a vox formula (V  = lwh) to make w that is the width as the subject of a formula.

Solution:

We will start with,

V = lwh

Dividing both the sides by h

V/h  = lwh/h

We get,

V/h = lw

Dividing both the sides by l

V/hl = lw/l

We get,

V/hl = w

Swapping the sides, we get

w = V/hl

Now, you can easily calculate the width of a box by applying the formula w= Vhl, where w is the width, v is the volume, and h is the height of the box.

2. Find the value of a, If 5a + 9a = 16 - 2a

Solution:

We have,

5a + 9a = 16 - 2a

5a + 9a + 2a = 16

16 a = 16

a = 16/16

a = 1

Fun Facts

• The first formula was introduced in between 1800 - 1600 BC.

• The first equation was written by Robert Recorde in his treatise “ The Whetstone of Witte” in 1557. In modern terms, the equation is represented by 14x + 15 = 71, and its solution is x = 4.