Variables and Constants: Definitions, Explanation, Examples

Variables and Constants: Algebra is a generalized form of arithmetic. In arithmetic, we use numbers like \(7, – 15,\frac{{12}}{{37}},0.075\) etc., each with a definite value, whereas in algebra, we use letters \(a, b, c, x, y, z,\) etc., along with numbers. For instance, \(10y, 2py, 3y+8z, 3x-6y+z\) and so on.

An algebraic expression is defined as the sum, product, difference, or quotient of constants and variables in algebra. This article will cover the definitions of constants and variables and learn their usage and importance while framing algebraic expressions or equations. 

Define Variables and Constants

Variables: In algebra, we come across two types of symbols, namely, constants and variables. A variable is a quantity that may change its value 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 it can represent many numbers. A symbol whose value changes with the situation is called a variable, like \(x, p, r, s, 6p,\) etc.

Constants: A symbol with a fixed numerical value in all situations is called a constant, for instance, \(10, – 11,0.025,\frac{{25}}{{11}}\) etc., 

In \(6p, 6\) is a constant and \(p\) a variable, but, together \(6p\) is a variable. As the value of \(p\) will change, the value of \(6p\) will also change accordingly. Similarly, \(5\) is a constant, and \(x\) is a variable, then each of \(5+x, 5-x, x÷5, 5÷x\) is a variable. So, we can conclude that every combination of a constant and a variable is always a variable.

Consider the following situation: 

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The taxi fare in Coimbatore is calculated as \(₹30\) times the number of kilometres travelled plus the base fare of \(₹90\). Thus, if the number of kilometres travelled is \(n\), the taxi fare will be \(₹(30n+90)\). As the number of kilometres travelled changes, the total fare changes accordingly. Since \(n\) can vary, it is called a variable, \(₹90\) is a fixed amount, and \(30n+90\) is called the algebraic expression.

Difference Between Variable and Constant

Let us consider a situation again.

Sonu worked at a game store for a day and earned \(₹300\). He was promised \(₹240\) as a fixed amount and earned \(₹20\) for every extra hour he put in.

Here in this situation, we can see that earned \(₹240\) is a fixed amount and earning per extra hour can vary. Let us take, per hour as \(x\) and thus, the equation can be \(₹240+20x=₹300.\)

Thus, what we understood about the variables and constants from the above-given situation is that 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 its value with the situation and a constant is a symbol whose value remains same.

Below given are the differences between variables and constants:

Algebraic Expressions and Terms

An algebraic expression is a combination of variables and constants connected by signs of fundamental operations, i.e., \(+, -, ×\) and \(\div\). For example, \(3x+2y-10\) is an algebraic expression. This expression is formed from variables \(x\) and \(y\), and constants \(2, 3\) and \(9\). The product of \(3\) and \(x\) is added to the product of \(2\) and \(y\), and from the sum \(3x+2y,\) the constant \(10\) is subtracted. 

Consider another algebraic expression \(3x-2y=9.\)

We can write it as \(3x+(-2y)=9\). In this algebraic expression, parts separated by plus signs are called terms. The expression \(3x+(-2y)=9\) consists of \(3\) terms. They are \(3x, -2y\) and \(9\). Thus, a term is a constant or a variable or a product or a quotient of constants and variables.

The numerical part, including the sign or symbol of any term, is called the variable’s coefficient. Thus, any factor or group of factors of a product is known as the coefficient of the remaining factors. In the above expression, \(3\) is the coefficient of \(x\), and \(-2\) is the coefficient of \(y.\) The term \(9\) with no variable is called a constant term. 

In a term, \(7pq,\) coefficient \(7\) is known as the numerical coefficient, and \(pq\) is known as the literal coefficient.

Let’s learn more deeply and become aware of few more terms.

a. \(2a + 3b\)
b. \(8{x^2}\)
c. \(4 x^{3}+3 x y-5 y^{2}+10 xy\)

The first algebraic expression has two terms, i.e., \(2a\) and \(3b.\) These are unlike terms as the variables \(a\) and \(b\) are different from each other. \(2\) is the coefficient of \(a,\) and \(3\) is the coefficient of \(b.\) Since there are two terms, and the algebraic expression is binomial. 

There is only one term in the second algebraic expression, i.e.,\(8{x^2}\) Therefore, it is a monomial. The numerical coefficient of \(8{x^2}\) is \(8.\) The third algebraic expression has \(4\) terms that are, \(4{x^3},3xy,5{y^2}\) and \(10xy.\) In this, \(3xy\) and \(10xy\) are like terms.

Finding the Value of an Algebraic Expression

If the variables in a given algebraic expression have defined values, substitute the values in the algebraic expression to find the value of the algebraic expression.

For example, if \(x=5,\) then \(10a+7=10×5+7=50+7=57\)           

Let us understand this better with the help of another example.

If \(a=9, b=3,\) find the value of \(5a+6b-10.\)
Substituting the values of \(a\) and \(b\) in the algebraic expressions, we get, 
\(5a+6b-10=5×9+6×3-10\)
\(=45+18-10\)
\(=63-10\)
\(=53\)

Solved Examples – Variables and Constants

Q.1. State the number of terms and the constant term in each of the following expressions.
a. \(2b – 3a + 5\)
b. \(3p + 9 – 4{p^2} + {p^3}\)
Ans: A term is a constant or a variable or a product or a quotient of constants and variables.
a. In the expression \(2b-3a+5,\) there are three terms, i.e., \(2b, -3a\) and \(5.\) The constant term in the given expression is \(5.\)
b. In the expression \(3p + 9 – 4{p^2} + {p^3}\) there are four terms, i.e., \(3p,9,4{p^2}\) and \({p^3}\) The constant term in the given expression is \(9.\)

Q.2. State the coefficient of \(x\) and \(y\) in the algebraic expression \(10x-10y+3.\)
Ans: The numerical part, including the sign or symbol of any term, is called the variable’s coefficient. In the given algebraic expression, \(10x-10y+3, x\) and \(y\) are variables here, and thus, the coefficient of \(x\) is \(10,\) and the coefficient of \(y\) is \(-10. \)

Q.3. Find the numerical coefficients of all the terms in the algebraic expression \(12{x^2} + 5xy – 2{x^2}y\)
Ans: There are three terms in the given algebraic expression. \(12\) is the numerical coefficient of \({x^2},5\) is the numerical coefficient of \(xy,\) and \(-2\) is the numerical coefficient of \({x^2}y.\)

Q.4. Explain how the following expression \(6xy+10\) is formed.
Ans: In \(6xy+10,\) we first obtain the product of \(x\) and \(y\) \((xy)\) multiplied by \(6\) and then add \(10\) to this product \((6xy).\) So, we get \(6xy+10.\)

Q.5. Separate the constants and the variables from each of the following: \( – 8y,10,5x,\frac{2}{3},\frac{4}{{3x}}, – \frac{{15}}{2},7{x^2},0,az,7p,\frac{{100}}{{11}}xyz\)
Ans: A symbol whose value changes with the situation is a variable, whereas a symbol with a fixed numerical value in all cases is called a constant. 
So, in the given set of expressions, the constants are \(10,\frac{2}{3}, – \frac{{15}}{2},0\) and the variables are, \( – 8y,5x,\quad \frac{4}{{3x}},7{x^2},az,7p,\frac{{100}}{{11}}xyz\)

Summary

In this article, we learned about the concept of the variables and the constants. We did the deep understanding by learning their definitions with the help of examples. We also learned the different terms related to algebraic expressions like terms, coefficient, binomial, monomial and polynomial. We also learned the difference between variables and constants. We also learnt how to find the value of algebraic expressions.

• (a) Variables: A variable is a non-fixed item whose value can change. The letters x, y, z, and so on are used to represent variables.
• (b) Constants: In a given context, a constant is an entity whose value is fixed. We don’t know what the constant’s value is, but we do know that it is constant. If the values of constants are unknown or not supplied, they are commonly represented by the letters a, b, c, p, q, and so on, and if the values are known, they are marked by particular numerical values (such as 3, etc.).

These are mostly symbols that serve as placeholders for values. Variables are typically represented by letters and do not have a set value. A variable’s value is unique and might differ from one circumstance to the next. In algebraic expressions, variables and constants are commonly utilised.

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Frequently Asked Questions (FAQs) – Variables and Constants

Q.1. How do you identify variables and constants?
Ans: If the value is fixed in an expression, then we identify them constants, whereas the terms that are in the form of alphabets or symbols, we identify as variables. Variables are represented using alphabets as their values are changing according to the situations.

Q.2. What are variables and constants? Give an example.
Ans:  A variable is a quantity that may change within the context of a mathematical problem. When we use a variable in a function, we know that it is not just one constant number, but it can represent many numbers. A symbol whose value changes with the situation is called a variable, such as \(x, p, r, s, 6p,\) etc., whereas a symbol with a fixed numerical value in all situations is called a constant, for instance, \(10, – 11,0.025,\frac{{25}}{{11}}\) etc., 

Q.3. How do you solve algebraic expressions with variables?
Ans: The solution of an algebraic expression is the value of the variables in that. We can solve algebraic expressions with variables by methods like the transposition method.

Q.4. What are variables and constants in algebraic expressions?
Ans: Consider the algebraic expression, \(3x+2y-10\) is an algebraic expression. This expression is formed from variables \(x\) and \(y,\) and constants \(2, 3\) and \(9.\)

Q.5. How do you find the variable in an expression?
Ans: The variables are values that change their value according to the situations. They are usually represented by alphabets or symbols.  In an expression, if we find such symbols or alphabets other than numbers, we can identify them as variables.

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