The basic operations that are being used in mathematics (especially in real number systems) are addition, subtraction, multiplication and so on. These operations can also be done on the algebraic expressions. Let us see them in detail.
Algebraic expressions (also known as algebraic equations) are defined as “the type of expressions”, in which it mainly consists of three parts namely,
Variables: Alphabetical letters which are associated with some integers.
Coefficients: The values (integers) that are associated with the variables.
Constants: Only the integers which are associated with the expressions.
Example: 25x + 16y+ 9, 2y2 – 3y, 2a + 3b + c, etc.
Note: All the Coefficients, Variables and Constants are associated with each other by various types of arithmetic operators.
The algebraic expressions are classified into three types, namely,
- Monomial or single variable expressions.
- Binomial or two variable expressions.
- Polynomial or multi variable expressions.
Mathematical Operations on Algebraic Expressions
Addition of Algebraic Expressions
Addition of algebraic expressions involves the following steps:
Step 1: Group all the like terms based on the coefficients, (i.e.) the coefficients consisting of the same variables are grouped together.
Step 2: All the grouped coefficient under the same variables should be performed ADDITION OPERATION (+) and should be written as a single coefficient term with its respective coefficient.
Step 3: Similarly, for all the like variables, the operations are needed to be done.
Step 4: For constants, add them and no need to include any variables.
Step 5: If there not exists any like term for any term, keep it as it is.
Let us take an example, consider we are adding two algebraic expressions,
5x + 6y + 7z and 6x – 7y + 3
Now let us group the like terms,
5x + 6x + 6y – 7y + 7z + 3
Now let us perform + operation on like terms and the result will be,
x coefficient: 6 + 5 = 11
y coefficient: 6 – 7 = -1
z coefficient: 7
So , the result is 11x – y + 7z + 3 .
Similarly, for another example,
9a + 7c and
19b + 4
The result is 9a + 19b + 7c + 4
Note: The addition operation can be done on any number of Algebraic Equations, not limited to only adding two expressions.
Subtraction of Algebraic Expressions
The subtraction of the algebraic expressions is very similar to the addition of the expressions. Though, it is recommended to use the Column Subtraction Method.
The following process is to be followed for the subtraction of algebraic expressions:
Step 1: Write the algebraic expressions line by line, one after other (newline).
Step 2: Group the like terms based on the variables, and they are needed to write the expressions in the same order of variables for all expressions.
Step 3: Toggle the sign of all the terms in the last expression that has been written (i.e.) Change + to – and – to +.
Step 4: Then perform the necessary operation for the expressions.
Step 5: It is recommended to solve a couple of expressions one by one if there are any number of algebraic equations to be solved.
Let us consider an example of two algebraic expressions to be solved,
5x – 7y + 9 and 8y + 7x – 1
Now let us group the like terms and solve them,
5x – 7y + 9
7x + 8y -1
Now let us toggle the sign of the last written expression by order,
5x – 7y + 9
(-)7x (-)8y (+)1
( ) represents the toggled sign.
Now the result would be,
x coefficient: 5 – 7 = -2
y coefficient: -7 – 8 = -15
constant: 9 + 1 = 10
Therefore, the result is -2x – 15y + 10
Now, let us consider another example,
6a – 7b – 1 and 4a – 6b – 1
The result is 2a – b (No constant because they cancel each other)
Multiplication of Algebraic Expressions
Multiplication is one of the frequent arithmetic operations that is been done on the algebraic expressions/equations. The following process can be followed to multiply a set of algebraic expressions,
Step 1: The multiplication is done by taking each and every term of the first expression and multiplying with each and every term of the second expression.
Step 2: While multiplying if the same variables exist, then add the powers (by PRODUCT RULE) and to be written as exponent with the variable.
Step 3: If different variables exist, then just write them as the product along with another variable.
Step 4: Then for multiplying coefficients, like signs will give a POSITIVE result and unlike signs will give a NEGATIVE result.
Step 5: Each and every term obtain after multiplication should be separated by their respective signs.
Let us see an example for multiplication of algebraic expressions,
5x + 6y and 9x + 9y
Here, 5x to be multiplied with second equation and 6y to be multiplied with second equation.
Now take the first term of the first expression and multiply with each and every term of the second expression by following the steps mentioned above.
(5x * 9x) + (5x * 9y) = 45x2 + 45xy
Similarly, for second term,
(6y * 9x) + (6y * 9y) = 54xy + 54y2
Now the result would be combining them,
45x2 + 45xy + 54xy + 54y2
Similarly, consider other examples,
4x and 6x – 8y + 2z
The result is 24x2 – 32xy + 8xz