Why we cannot add unlike terms?

The basic concept of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are termed here as variables. this expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is termed a coefficient. An idea of expressing numbers using letters or alphabets without specifying their actual values is defined as an algebraic expression.

Algebraic Expression

An expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, etc. is termed an algebraic expression. These Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable. A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as an algebraic expression (or) a variable expression. Examples: 2x + 4y – 7, 3x – 10, etc.

The above expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three terms is termed as an expression. Unlike the algebraic equation, It has no sides or ‘equals to’ sign.

Types of Algebraic expression

There are three types of algebraic expressions based on the number of terms present in them. They are monomial algebraic expressions, binomial algebraic expressions, and polynomial algebraic expressions.

• Monomial Expression: An expression that has only one term is termed a Monomial expression. Examples of monomial expressions include 5x⁴, 2xy, 2x, 8y, etc.
• Binomial Expression: An algebraic expression which is having two terms and unlike are termed as a binomial expression.  Examples of binomial include 5xy + 8, xyz + x², etc.
• Polynomial Expression: An expression that has more than one term with non-negative integral exponents of a variable is termed a polynomial expression. Examples of polynomial expression include ax + by + ca, 3x³ + 5x + 3, etc.

Some Other Types of Expression

Apart from monomial, binomial, and polynomial types of expressions, there are other types of expressions as well that are numeric expressions, variable expressions.

• Numeric Expression: An expression that consists of only numbers and operations, but never includes any variable is termed a numeric expression. Some of the examples of numeric expressions are 11 + 5, 14 ÷ 2, etc.
• Variable Expression: An expression that contains variables along with numbers and operations to define an expression is termed A variable expression. Some examples of a variable expression include 5x + y, 4ab + 33, etc.

Some algebraic formulae

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. (a + b)(a – b) = a² – b²
4. (x + a)(x + b) = x² + x(a + b) + ab
5. (a + b)³ = a³ + b³ + 3ab(a + b)
6. (a – b)³ = a³ – b³ – 3ab(a – b)
7. a³ – b³ = (a – b)(a² + ab + b²)
8. a³ + b³ = (a + b)(a² – ab + b²)

Why we cannot add unlike terms?

Answer:

Terms that are having different variables or terms having variables with different exponent power for them are called Unlike terms. Example: 5z & 16x and 7x² & 7x³. Here, 5z and 16x are called unlike terms because they have different coefficients of z and x.

The terms with the same variable with different exponents or different variable with same exponents are called Unlike terms. Only like terms can be added or subtracted.

The sum of one or more like terms is a single like term whereas the two unlike terms cannot be added together to get a single term.

Let’s take a look at this with an example,

If 3x² + 3xy + 4x + 7 is an algebraic expression.

Then, 3x², 3xy, 4x, and 7 are the Terms

Coefficient of the term: 3 is the coefficient of x2

Constant term: 7

Variables: Here x, y are variables

Factors of a term: If 3xy is a term, then its factors are 3, x, and y.

Like and Unlike Terms: Example of like and unlike terms:

Like terms: 2x and 3x

Unlike terms: 3x and 4y

Sample Problems

Question 1: Add 2z & 16x.

Answer:

Here the terms present are 2z & 16x,

2z + 16x is a term but this cannot be added because both have different variables and are unlike terms.

Question 2: Identity like terms from the following and add?

3zy²x, 7xy²z, 3xz²y, 4x²yz

Solution:

Like Terms: 3zy²x, 3xy²z

Now add 3zy²x, 3xy²z

= 3zy²x + 3xy²z

=  6xy²z 

Hence only like terms can be added together

Question 3: Add (3x² – 5xy + 7 + z³) & (3x² + 4xy – 6 + 2z³).

Solution:

There are, (3x² – 5xy + 7 + z3) & (3×2 + 4xy – 6 + 2z3)

Add like terms together,

= (3x² – 5xy + 7 + z³) + (3x² + 4xy – 6 + 2z³)

= 3x² – 5xy + 7 + z³ + 3x² + 4xy – 6 + 2z³

= 3x² + 3x² – 5xy + 4xy + z³ + 2z³ + 7 – 6

= 6×2 – xy + 3z³ + 1

Question 4:  Simplify (3x – 5) – (6x + 1)

Solution:

Given that, (3x – 5) – (6x + 1)

• Step 1: Remove parentheses and apply the signs carefully.

= 3x – 5 – 6x – 1

• Step 2: Bring like terms together

= 3x – 6x – 5 – 1

• Step 3: Now add or subtract the like terms

= -3x – 6

= -3(x + 2)

So the final result is -3(x + 2)

Question 5: Identify and Add like terms together?

5x, 6x², 89xy², 7x⁴, 3x, 34xy²

Answer:

Here Like terms are : 5x , 3x , 89xy , 34 xy 

Now add: 5x + 3x = 8x 

89xy² + 34xy² = 123xy²