This article will explore the Like and Unlike Terms and enhance the knowledge about algebraic expressions. An algebraic expression is a combination of variables and constants connected by the signs of fundamental operations, i.e., \(+,-, x\) and \(\div\). For example, \(3 x+2 y-9\) is an algebraic expression.

In an algebraic expression, the parts separated by plus or minus signs are called terms. For instance, the expression \(2 x+5 y+9\) consists of three terms. They are \(2 x, 5 y\) and \(9\). The numerical part, including the sign of a term, is called the numerical coefficient of the variable. Terms can be of two types, namely, like terms and unlike terms. Continue reading to know more.

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Like and Unlike Terms in Algebra

A term is a constant or a variable or a product or a quotient of constants and variables. For example, \(4\) is a term, which is a constant. \(y\) is a term with a variable. \(3 z\) is a term, which is the product of a constant and a variable.

Consider an algebraic expression \(7 x-5 y=9\)

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We can write it as \(7 x+(-5 y)=9\). In this algebraic expression, parts separated by plus signs are called terms. The expression \(7 x+(-5 y)=9\) consists of \(3\) terms. They are \(7 x,-5 y\) and \(9\).

The numerical part, including the sign or symbol of any term, is called the variable’s numerical coefficient. Thus, any factor or group of factors of a product is known as the coefficient of the remaining factors. In the above expression, \(7\) is the coefficient of \(x\), and \(-5\) is the coefficient of \(y\). The term \(9\) with no variable is called a constant term.

Thus, we can say that when two terms have identical variable parts, i.e., they consist of the same variable(s) having the same exponent(s), they are called like terms. For example, \(4 x y\) and \(-10 x y\) are like terms as they have identical variable parts, \(x y\). All constant terms are like terms. When two terms are not like terms, they are called, unlike terms.

Like Terms	Unlike Terms
\(4x, 11x\)	\(7a, -10b\)
\(3 x^{2}, 9 x^{2}\)	\(4 x^{2}, 7 y^{2}\)
\(\frac{-4 p q^{2}}{11}, \frac{11 p q^{2}}{13}\)	\(\frac{-5 p q^{2}}{19}, \frac{11 p^{2} q}{13}\)
\(12, -3\)	\(16p, -30\)

Like and Unlike Terms Examples

We have already learnt that the terms having the same literal coefficients are called like terms, and they may differ only in their numeral coefficients, whereas the terms that do not have the same literal coefficients are called, unlike terms.

Let us look at some examples and separate the like and unlike terms.

Example 1: Separately group the like and unlike terms.

a) \(4 x, 3 y z,-x, \frac{3}{5} x, 14 x y z, \frac{-15}{3} x, 10 x, 3 a b c,-a b\)
b) \(\frac{-5}{3} a b c, 4 a b,-2 b c, a b c, 2 b c a, 5 c a b, \frac{-15}{16} x y z,-x y\)

Solution: a) From the given set of terms, \(4 x, 3 y z,-x, \frac{3}{5} x, 14 x y z, \frac{-15}{3} x, 10 x, 3 a b c,-a b\), the like terms and unlike terms can be separated as

Like Terms	Unlike Terms
\(4 x,-x, \frac{3}{5} x, \frac{-15}{2} x, 10 x\)	\(3 y z, 14 x y z,, 3 a b c,-a b\)

c) From the given set of terms, \(\frac{-5}{3} a b c, 4 a b,-2 b c, a b c, 2 b c a, 5 c a b, \frac{-15}{16} x y z,-x y\), the like terms and unlike terms can be separated as

Like Terms	Unlike Terms
\(\frac{-5}{3} a b c, a b c, 2 b c a, 5 c a b\),	\(4 a b,-2 b c,, \frac{-15}{16} x y z,-x y\)

\(ABC, bca\) & \(cab\) are like terms as the combination of variables are the same even though the order of variables is different.

Let us look at another example to understand better about the like and unlike terms.

Example 2: Identify the pairs of like and unlike terms.

a) \(9 x, 16 y\)
b) \(-10 a b, 9 b a\)
c) \(21 x,-21 x\)
d) \(16 p q r^{2}, 16 p q^{2} r\)
e) \(-x y z, 2 x y\)

Solution: Pairs with the same variables will be the like terms, and pairs with the different variables will be unlike terms.

So, the pairs \(21 x,-21 x\) and \(-10 a b, 9 b a\) are having like terms. Other pairs have, unlike terms.

Difference Between Like and Unlike Terms

By now, we are already aware of the like terms and unlike terms. Let us look at the difference between them.

The difference between like terms and unlike terms is summarized below.

Like terms	Unlike terms
Have identical variable parts, i.e., they consist of the same variable(s) having the same exponent(s)	Have different variable parts, i.e., they consist of different variable(s)or have different exponent(s)
Like terms can be further simplified by combining them	Unlike terms cannot be further simplified by combining them
Like terms can be added and subtracted together	Unlike terms cannot be added and subtracted together
For example, \(4x\) and \(11x\) are like terms as they have the same variable part	For example, \(7a\) and \(-10b\) are unlike terms as they have a different variable part

Classification of Algebraic Expressions Based on Terms

Monomial: An algebraic expression having only one term is called a monomial. For example, \(4 x,-x y, \frac{-5}{3} a b c\), etc., are monomials.

Binomial: An algebraic expression that contains two unlike terms is called a binomial. For example, \(x+y, 4 p+2 z, 3 x^{2}-y^{2}\) etc., are binomials.

Trinomial: An algebraic expression having three unlike terms is called a trinomial. For example, \(a+4 b+2 z, x-p q+y z\), etc., are trinomials.

Polynomial: All monomials, binomials, trinomials and all other expressions having any number of finite terms with exponents as whole numbers are called polynomials. No term of polynomial contains a negative exponent or any variable in the denominator. For example, \(4 x^{3}+3 x^{2}+9 x-1,9 a^{2}+5\), etc., are polynomials.

Addition and Subtraction of Like and Unlike Terms

In mathematics, addition, subtraction, multiplication, and division are the four fundamental operations. Let us learn them to apply on like terms as well as unlike terms.

Addition and Subtraction of Like terms

The sum of like terms is a single term whose coefficient is equal to the sum of the coefficients of the given terms. For subtraction of like terms, the rules are the same as those for the subtraction of integers.

Let us understand this with the help of examples.

Example 1: Add \(4 x y, 6 x y\), and \(10 x y\)

Solution: \(4 x y+6 x y+10 x y=(4+6+10) x y=20 x y\).

Example 2: Subtract \(-4 a\) from \(-10 a\)

Solution: \(-10 a-(-4 a)=-10 a+4 a=-6 a\)

Addition and Subtraction of Unlike terms

As we have seen above in the examples, the sum and difference of two or more like terms is a single term, but two unlike terms cannot be added or subtracted together. For example, the unlike terms \(2 x y\) and \(4 y z\) cannot be added or subtracted together to form a single term. Instead, they can be connected by the sign of addition or subtraction and to get the result in the form of \(2 x y+4 y z\) or \(2 x y-4 y z\).

Solved Example Problems on Like and Unlike Terms

Q.1. Identify the pair of like terms and unlike terms.
a) \(3 a^{2} b,-a^{2} b\) b) \(-4 a b c, 4 b a c\) c) \(3 x y^{2},-5 x^{2} y^{2}\) d) \(8 a b c^{2}, \frac{11}{2} a^{2} b c\)
Ans: We know that the terms with the same literal coefficients are called like terms, whereas those with the same literal coefficients are called, unlike terms.
Thus, \(\left(3 a^{2} b,-a^{2} b\right)\) having the same literal coefficient as \(a^{2} b\) and \((-4 a b c, 4 b a c)\) having the same literal coefficient as \(a b c\) are like terms. Whereas \(\left(3 x y^{2},-5 x^{2} y^{2}\right)\) and \(\left(8 a b c^{2}, \frac{11}{2} a^{2} b c\right)\) have different literal coefficients, and so they are unlike terms.

Q.2. In the algebraic expression, identify the unlike terms \(-32 x^{2}-7 y+4 x^{2}+43 x^{2}-5 x y+12 x\)
Ans: The given expression is \(-32 x^{2}-7 y+4 x^{2}+43 x^{2}-5 x y+12 x\).
Therefore, the, unlike terms, are \( – 7y,\, – 5xy\,\& \, + 12x\) as the variable and coefficients are different from each other. \(-32 x^{2}, 4 x^{2} \,\&\, 43 x^{2}\) are like terms.

Q.3. Group the like and unlike terms separately from the given below.
\(13 p, 12 x y,-5 p, \frac{11}{2} a^{2}, \frac{-6}{7} x^{2}, p, 5 p x, 12 a p, 100 p\), and \(2 p\)
Ans: The separation of like and unlike terms is shown below in the table.

Like Terms	Unlike Terms
\(13 p,-5 p, p, 100 p, 2 p\)	\(12 x y, \frac{11}{2} a^{2}, \frac{-6}{7} x^{2}, 5 p x, 12 a p\)

Q.4. Evaluate: \(8 a-3 a+12 a+13 a-6 a\)
Ans: In the given expression, \(8 a-3 a+12 a+13 a-6 a\), all are like terms and thus can be further simplified.
Thus,
\(8 a-3 a+12 a+13 a-6 a=(8-3+12+13-6) a\)
\(=24 a\)
Hence, the required answer is \(24 a\).

Q.5. Simplify: \(13 p q r-2 p+4 p-6 p q r+5 p q r\)
Ans: To simplify \(13 p q r-2 p+4 p-6 p q r+5 p q r\) gather the like terms together.
Thus, after rearranging the terms, we get, \(13 p q r-6 p q r+5 p q r-2 p+4 p\)
\(=(13-6+5) p q r+(-2+4) p\)
\(=12 p q r+2 p\)
Hence, the required answer is \(12 p q r+2 p\).

Summary

In this article, we did a discussion about the like terms and unlike terms. Firstly, we understood the terms in an algebraic expression and later learnt that terms could be divided into like and unlike terms, and later, we learnt more about like and unlike terms. In addition to this, we also learnt to perform addition or subtraction on like and unlike terms.

FAQs

Q.1. Define like and unlike terms with examples.
Ans: Two terms with identical variable parts, i.e., the same variable(s) having the same exponent(s), are called like terms. For example, \(4 x y\) and \(-10 x y\) are like terms as they have identical variable parts, \(x y\). All constant terms are like terms. When two terms are not like terms, they are called, unlike terms. For example, \(3 y z, 14 x y z, 3 a b c,-a b\) are unlike terms with different variable parts.

Q.2. How do separate like and unlike terms?
Ans: Look at the variable part of the terms, and then it becomes easy to separate like and unlike terms. As like terms will have the same variable part and unlike terms will have the different variable part.

Q.3. What are the differences between like terms and unlike terms?
Ans: The differences between like and unlike terms are summarized below.

Like terms	Unlike terms
Have identical variable parts, i.e., they consist of the same variable(s) having the same exponent(s),	Have different variable parts, i.e., they consist of different variable(s)or have different exponent(s),
Like terms can be further simplified by combining them	Unlike terms cannot be further simplified by combining them
Like terms can be added and subtracted together	Unlike terms cannot be added and subtracted together.

Q.4. How do we find like and unlike terms?
Ans: In an algebraic expression, if the variables are the same despite different coefficients and the exponents being the same, those terms are known as like terms. Whereas, if the expression consists of two different variables or different exponents or coefficients, those expressions are known as, unlike terms.

Q.5. Can we combine unlike terms together?
Ans: No, unlike terms in an algebraic expression, can neither be combined. Because the unlike terms contain two different variables and thus cannot be combined.

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