Contents

# Variable|Definition & Meaning

## Definition

A variable is something whose value is unknown. We usually represent variables using letters from the English alphabet (usually the last few, like x, y, etc.). For example, in x + 1 = 3, x is a variable. This equation is only true if x = 2. However, there are equations (x + y = 3) in which multiple values of the same variable are valid, so we just call them variables in general.

Figure 1 illustrates the difference between the coefficient, Variable and Constant.

Single letters are used to designate geometrical points and forms in **prehistoric writings** like Euclid’s Elements. In the Brhmasphuasiddhnta from the 7th century, Brahmagupta employed several colours to symbolize the unknowns in algebraic equations. Equations of **Several Colours** is the title of one chapter in this book.

The concept of expressing known and unknown numbers by letters now referred to **as variables**, and computing with them as if they were numbers in order to reach the answer by a simple replacement was first proposed by **FranÃ§ois ViÃ¨te** towards the end of the 16th century. Consonants were used for known values and vowels for unknowns according to **ViÃ¨te’s convention.**

Typically, variables are identified by a single letter, either **capitalized or lowercase**, most frequently from the Latin alphabet and less frequently from the Greek. The subscript may come after the letter. **Pure mathematics** variable names that have been influenced **by computer technology** can include many letters and numbers in them.

Letters at the start of the alphabet, such as a, b, and c, are frequently used for known **values and parameters**, whereas the last few letters of the alphabet, such as x, y, and z, are frequently used **for unknowns and variables of functions**, according to RenÃ© Descartes (1596-1650). Variables and constants are typically set in italic font in **printed mathematics.**

**Equation**

Two things are equal, according to an equation. It will include the equals **symbol “=”.** A term might be a single number, a variable, or a combination **of numbers and variables multiplied**. A set of terms (the terms are separated by + or signs) is referred to as an expression. An example of an equation is illustrated in figure 2.

**Exponents**

The exponent (such as 2 in $x^2$) specifies how many times to use the value in the multiplication.

Figure 3 illustrates the exponent.

**Specific Names for the Variables**

A variable that has to be solved in an equation is an unknown. **An indeterminate** is a symbol, sometimes known as a variable, that may be found in a formal power series or a polynomial. In the **ring of formal power series** or the polynomial ring, **an indeterminate is a constant** rather than a variable.

However, due to the close connection between **polynomials or power series** and the functions they describe, many writers view indeterminates as a **particular category of variables. **A parameter is a quantity, generally a number, that is included in a problem’s input and is consistent during the whole solution process.

For instance, in mechanics, the size and mass of a **solid entity** are criteria for the investigation of its motion. The term **“parameter”** has a distinct meaning in computer science and refers to a function argument. There are bound and unbound variables. In **probability theory** and its applications, a random variable is a **type of variable.**

**Types of Variable **

In calculus and physics, and other **scientific** **applications,** it is **fairly** common to consider **variables** **such** **as** **y,** whose possible values **depend** on the **values** **of** another **variable** **such** **as** x. **Mathematically** **speaking,** the dependent variable y represents the value of a function of x.

To simplify **expressions,** it often **makes** **sense** to use the same **notation** for the dependent variable y and the function **that** **maps** x **to** y. For example, the state of a physical system depends on measurable quantities such as pressure, temperature, **and** spatial **position.**

**All** **of** these quantities **change** **as** the system **evolves.** **That** is, they are **functions** of time. In the formulas **that** **describe** the system, these **variables** **are** **implicitly** **considered** **functions** **of** **time** **because** **they** are represented by **time-dependent** **variables.**

The variables may be divided into two groups, including

- Dependent Variables
- Independent Variables

**Dependent Variables **

The term “**dependent variable**” refers to a variable whose evaluation of another variable in its context determines the quality of the dependent variable. That is, it is consistently claimed that the **estimation of the word variable** depends on the free variable of the mathematical condition.

Take the formula y = 4x + 3 as an example. In this scenario, **alterations to the estimation** of the variable ‘x’ cause changes in the estimation of the variable ‘y’. In this way, the variable “y” is referred to as **a dependent variable**. Some of the instances that include dependent variables are discussed in the point of interest below, **along with their solutions.**

**Independent Variables**

An **independent variable** in an algebraic equation is a variable whose values are unaffected by changes. In an **algebraic equation** with two variables (x and y), if any value of x is related to any other value of y, then it is said that the value of **y is a function** of the value of the independent variable (x), which is known as the x value, and the value of y is known as the **dependent variable.**

**Applications of Variables**

Variables are utilized not only in mathematics but also in many other disciplines for a variety of reasons, including programming, research, science, and statistics

**Some Examples of Variables**

**Example 1**

Solve the following

a) Find x in the equation 6x + 5 = 17.

b) Find y in the equation 3y + 15 = 21.

c) Find the factors of the equation x^{2}+6x+9.

**Solution**

a) As this is an equation, both sides are equal, as represented by the equal sign. To find a value of x that satisfies the right-hand side (RHS) and left-hand side (LHS) of the equation, we will do some algebraic manipulations:

6x + 5 = 17

6x = 17 â€“ 5

6x = 12

x = $\mathsf{\dfrac{12}{6}}$

**x = 2**

If we put x is equal to two, then 6(2) + 5 is equal to 17.

b) As this is an equation, both sides are equal, which is why there is an equal sign. In order to find y, which satisfies the right-hand side (RHS) and left-hand side (LHS) of the equation, we will do some algebraic manipulations:

3y + 15 = 21

3y = 21 â€“15

3y = 6

y = $\mathsf{\dfrac{6}{3}}$

**y = 2**

if we put x is equal to two then 6(2)+5 is equal to 17.

c) In order to find the factors x^{2} + 6x + 9, we have to factorize the equation, which can be done as follows:

= x^{2} + 6x + 9

x^{2} + 3x + 3x + 9

x(x + 3) + 3(x + 3)

(x + 3)^{2}

x is equal to 3 and -3.

**Example 2**

Solve the following

a) Find y in equation 5y + 15 = 17.

b) Find z in equation 3z + 15 = 21

**Solution**

a) As this is an equation, both sides must be equal for the equation to hold true. In order to find y, which satisfies the right-hand side (RHS) and left-hand side (LHS) of the equation, we will do some algebraic manipulations:

5y + 15 = 17

5y = 17 â€“ 15

5y = 3

y = $\mathsf{\dfrac{3}{5}}$

y = 0.6

if we put x is equal to two then 5(0.6)+15 is equal to 17.

b) As this is an equation, so both sides are equal, which is why there is an equal sign. In order to find a value for z that satisfies the right-hand side (RHS) and the left-hand side (LHS) of the equation, we will do some algebraic manipulations:

3z + 15 = 21

3z = 21 â€“15

3z = 6

z = $\mathsf{\dfrac{6}{3}}$

z = 2

if we put x is equal to two then 3(2)+15 is equal to 21.

*All images were created using GeoGebra.*