In algebra, we come across two types of symbols, namely, constants and variables. A symbol with a fixed numerical value in all situations is called a constant. For example, \(7,\,15,\, – 9,\,\frac{8}{9}~\) etc., whereas a symbol whose value changes with the situation is called a variable such as \(z, xy, y\), etc.

Do you know there are different types of variables as well? And are you aware that the constant and variable combined form an algebraic expression connected by signs of fundamental operations, i.e., \(+, -, ×\) and \( \div \). In this article, let’s study the concept of variables in Maths.

Definition of Variables

A variable is a quantity that may change within the context of a mathematical problem. When a variable is used in a function, we know that it is not just one constant number, but that can represent many numbers. Letters like \(x, y, z\) are the generic type of variables and are used most of the time, but at times we will choose a letter that reminds us of the quantity it represents, such as a for acceleration, \(t\) for time etc.

Check the figure given below for a better understanding.

What is Meant by Variable?

Sample this, the taxi fare in Bangalore is calculated as \(₹15\) times the number of kilometres travelled plus the base fare of \(₹70\). Thus, if the number of kilometres travelled is n, the taxi fare will be \(₹15n+70\). As the number of kilometres travelled changes, the total fare changes accordingly. Since n can vary, it is called a variable and \(15n+70\) is called the algebraic expression.

Thus, what we understood about the variable from the above-given situation can be generalized as ‘a variable is a symbol or an alphabet representing an unspecified term, or which works as a placeholder for an algebraic expression that may vary or change with the situation.

Have a look at the image given below for a better understanding

What is Variable in Statistics?

We have learned about the variables and their usage in algebraic expressions. The term variable is used in statistics also. In statistics, a variable at times is called a data item. It represents the number or, let’s say, characteristics that can be counted or measured. For example, age, income, capital expenditure, gender, and class grades are examples of variables in statistics.

Uses of Variables

We come across equations with variables such as \(x+7=21\) and \(y=4x-7z\), etc. However, one must understand that variables can be used for various purposes apart from mathematical problems.

Assume you went out to eat pizza at your favourite restaurant. They sell double cheese Margherita for \(₹100\). Now, you want to get as many possibilities as you can, but you have only \(₹320\). Along with double cheese Margherita, you also want a drink that costs \(₹20\). Now, you need to figure out how much double cheese Margherita you want? Thus, here the number of double cheese Margherita is your variable. So, let’s call it \(m\).

Now, each double cheese Margherita costs \(₹100\), so \(100m\) is the cost of double cheese Margherita multiplied by the number of double cheese Margherita. \(100m+20\) is the cost of double cheese Margherita and the drink, which equals \(₹320\), the money you have. Therefore, the equation comes to \(100m+20=320\).

Did you notice, you just used the variable to help you solve the problem and helped yourself to figure out the number of double cheese Margherita you can buy with the amount you have.

Now let’s solve and find out the number of double cheese Margherita you can have.

\(100m+20=320\)
\(100m=320-20\)
\(100m=300\)
\(m=3\)

And, thus, you can have \(3\) double cheese Margherita along with a drink of \(₹20\), which equals \(₹320\), with the money you have.

Types of Variables

We are now thorough with the definition of variables. Variables can be mainly classified into \(2\) types.

1. Dependent variables
2. Independent variables

To understand, let us start with the image given below,

As we can see on the left side of the given image, the amount of water is a type of independent variable. Now, have a look at the other half of the image; we can see that the growth of the plants is dependent on the amount of water given to the plants to nurture. And, since the growth of plants is dependent on the water provided, so, the growth of plants is a type of dependent variable.

1. Dependent variable: A dependent variable is a type of variable that depends on the value of some other number. It can also be characterised as a variable whose output solely depends on the estimation of another variable. In short, the dependent variable is the output of a function. And, thus, the outcome of the dependent variable changes every time if there is a change in the value of an independent variable.

For instance, in the equation \(x=2+5y, x\) value is totally dependent on the value of the function \(2+5y\). Hence, \(x\) is a dependent variable.

2. Independent variable: According to the algebraic expressions, the independent variable (as they have their own value), does not depend on any other values. It is also known as the input of a function.

For instance, the equation \(x=10-2z, z\) is an independent variable as the value of \(z\) is not affected by any other values.

Apart from dependent and independent variables, there are three more types of variables.

1. Random variable: A random variable is a variable whose value depends on the outcomes of a random experiment.
2. Continuous variable: A continuous variable is defined as a variable that can take an uncountable set of values.
3. Categorical variable: A categorical variable is a variable that can take on a fixed number of possible values.

Applications of Variables

As mentioned above, variables are used in various other fields apart from Maths. Some of the fields where variables are significantly used are as follows:

1. Programming
2. Research
3. Science
4. Statistics

Solved Examples on Variables

Q.1. Solve the equation \(\mathbf4\boldsymbol x\boldsymbol+\mathbf7\boldsymbol=\mathbf{19}\).
Ans: The given equation is \(4x+7=19\).
\(4x=19-7\)
\(4x=12\)
\(x=3\)

Q.2. Find the value of the variable \(\boldsymbol y\) for the equation \(\boldsymbol y\boldsymbol=\mathbf5\boldsymbol\times\mathbf2\) when \(\boldsymbol x\boldsymbol=\mathbf4\).
Ans : Given equation: \(y = 5{x^2}\). Here \(x\) is an independent variable and \(y\) is a dependent variable.
Now, substituting \(x = 4\) in the given equation, we get
\(y = 5 \times {4^2}\)
\(y = 5×(16)\)
\(y = 80\)

Therefore, the value of \(y\) is \(80\), when \(x = 4\).

Q.3. When \(\boldsymbol x\boldsymbol=\mathbf4\) and \(\boldsymbol y\boldsymbol=\mathbf1\) , find the value of the \(\boldsymbol x^{\mathbf2}\boldsymbol y\boldsymbol–\mathbf3\boldsymbol x\boldsymbol y\boldsymbol+\mathbf5\).
Ans: When \(x=4\) and \(y=1\),
\({x^2}y – 3xy + 5 = {(4)^2} \times 1 – 3 \times 4 \times 1 + 5\)
\(=16-12+5\)
\(=9 \)

Hence, the value of the \({x^2}y – 3xy + 5\) when \(x=4\) & \(y=1\) is \(9\).

Q.4. The area, \(\boldsymbol A\boldsymbol\;\boldsymbol c\boldsymbol m^{\mathbf2}\), of a rectangle of length, \(\boldsymbol l\boldsymbol\;\boldsymbol c\boldsymbol m\) and breadth \(\boldsymbol b\boldsymbol\;\boldsymbol c\boldsymbol m\) is given by the formula \(\boldsymbol A\boldsymbol=\boldsymbol l\boldsymbol b\). Find the area of the rectangle whose length is \({\rm{12\;cm}}\) and breadth is \({\rm{7\;cm}}\).
Ans: Given,\(A=lb\)
\(l = 12\;{\rm{cm}}\) and \(b = 7\;{\rm{cm}}\)
\(A = 12\;{\rm{cm}} \times 7\;{\rm{cm}}\)
\( = 84\;{\rm{c}}{{\rm{m}}^2}\)

Therefore, the area of the rectangle is \( = 84\;{\rm{c}}{{\rm{m}}^2}\).

Q.5. Given the formula \(\boldsymbol S\boldsymbol=\frac{\mathbf1}{\mathbf2}\boldsymbol n\boldsymbol(\boldsymbol n\boldsymbol+\mathbf1\boldsymbol)\). Find the value of \(S\) when \(\boldsymbol n\boldsymbol=\mathbf{10}\).
Ans: When \(n=10\), we must find the value of \(S = \frac{1}{2}n(n + 1)\).
\(S = \frac{1}{2}n(n + 1) = \frac{1}{2} \times 10 \times (10 + 1)\)
\(=55\)

Note: The formula \(S = \frac{1}{2}n(n + 1)\) relates the variables \(n\) and \(S\). When the value of \(n\) changes, the value of \(S\) also changes.

Summary

In this article, we learned about the need for variables in real life, and in addition to that, we understood the definition of a variable with the help of examples. In algebra, we come across two types of symbols, namely, constants and variables. A symbol with a fixed numerical value in all situations is called a constant. The constant and variable combined form an algebraic expression connected by signs of fundamental operations, i.e., \(+, -, ×\) and \( \div \). We also learned about the two main types of variables and understood their meaning as well.

FAQs on Variables

Q.1. What is the definition of a variable?
Ans: A variable is a quantity that may change with the context of a mathematical problem. When a variable is used in a function, it is not just one constant number, but it can represent many numbers.

Q.2. What are the types of variables?
Ans: The types of variables are as follows:
1. Dependent variable
2. Independent variable
3. Random Variable
4. Continuous Variable
5. Categorical Variable

Q.3. What is variable and example?
Ans: A variable is a symbol or an alphabet representing an unspecified term or which works as a placeholder for an algebraic expression that may vary or change with the situation. In the equation, \(v=u+at, v, u, a, t\) are all variables.

Q.4. Define dependent variable.
Ans: Dependent variable is defined as a type of variable that depends on the value of some other number.

Q.5. What is an independent variable?
Ans: The independent variable is defined as a type of variable that does not depend on any other values.

Now that you are provided with all the necessary information about variables in Mathematics and we hope this article is helpful to you. If you have any queries on this page, post your comments in the comment box below and we will get back to you as soon as possible.

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