Defining an Algebraic Expression
An algebraic expression is a relationship between variables, constants and coefficients, and math operations like addition, subtraction, multiplication and division. The unknown value in an algebraic expression is called a variable. Variables are represented by letters such as a,b,e,f,x and y. There are expressions with a number, a fraction or even a decimal associated with variables. This value linked to a variable is called a coefficient. We also find numbers, fractions or even decimals in an algebraic expression without being linked to variables. These values are called constants.
A few examples of algebraic expressions are:
- \(4d^{2}+7xy+3ab+3\)
- \(c+6fg+12x\)
- \(6g+2g^{2}+5gh\)
What are ‘Like Terms’?
Now that we know the definition of algebraic expressions, understanding what terms are will be easy. An algebraic expression can consist of a number, a variable, product of a number and variable and product of two or more variables. Each of these parts of an expression are terms and an expression can have one or more such terms.
Let us consider a few expressions to understand how terms are a part of an algebraic expression:
Example 1:
\(3x^{2}+4xy+15\)
Variables: x, y
Coefficients: 3, 4
Constant: 15
Hence, the terms in this expression are \(3x^{2}, 4xy, 15\)
Example 2:
\(4xy+5y^{2}+7xy -12\)
Variables: x,y
Coefficients: 4,5,7
Constant: -12
Hence, the terms in this expression are \(4xy,5y^{2},7xy ,12\)
From these expressions we can observe that an algebraic expression is a collection of terms.
In the second example we can observe that the terms 4xy and 7xy have the same variables, ‘x’ and ‘y’. Even though the coefficients of these terms are the same, the variables in these two terms are the same. Hence, these two terms 4xy and 7xy with the same variables are called like terms.
Why do we Combine like Terms?
When there are like terms in an expression separated by mathematical operators such as addition and subtraction, we can group the like terms in order to make the calculation easier.
Let’s consider an expression:
\(3x^{2}+4xy+8wx+12x^{2}+12xy+3wx+6y\)
Let’s first identify the like terms in this expression:
Terms linked to the variable \(‘x^{2}-3x^{2}, 12x^{2}\)
Terms linked to the variable \(‘xy-4xy, 12xy\)
Terms linked to the variable \(‘wx-8wx, 3wx\)
Terms linked to the variable \(‘y-6y\)
Now that the like terms have been identified, we just group them and add them up to obtain a new expression.
This gives us,
\(3x^{2}+12x^{2}+4xy+12xy+8wx+3wx+6y\) [‘Grouping’ like terms]
\(\Rightarrow 15x^{2}+16xy+11wx+6y\) [Adding the like terms to obtain new expression]
So, this example shows us that grouping or combining like terms helps in easier and accurate calculations and this leads to simplifying algebraic expressions.
Rapid Recall
Solved Examples
Example 1:
Ryan has 3c + 14y number of marbles and Jenny has 14y + 3c number of marbles. Find the total number of marbles.
Solution:
Combine the like terms in each expression provided.
Number of marbles Ryan has = 3c+ 14y
Number of marbles Jenny has = 14y + 3c
Total number of marbles = (3c+ 14y) + (14y + 3c) [Add the expressions]
= (3c + 3c) + (14y + 14y) [Combine like terms]
= 6c + 28y [Find the sum]
Hence Ryan and Jenny have a total number of 6c + 28y marbles.
Example 2:
State whether the like terms can be combined in the following expressions.
Expression 1: \(2x^{2}+5y+17xy+21\)
Expression 2: \(5tx^{2}+6tu+7t^{2}+15tu\)
Expression 3: \(7fg+8jk+4lm+4fg+4jk\)
Expression 4: \(12rx+3tx+4gh+3ty\)
Solution:
Terms in Expression 1:
\(2x^{2},5y,17xy,21\) – None of these terms have any common variables.
Hence, the terms in this expression cannot be combined.
Terms in Expression 2:
\(5t^{2},6tu,7t^{2},15tu\) – In this expression, \(5t^{2}\) and \(7t^{2}\) can be combined to give \(12t^{2}\)
6tu and 15tu can be combined to give 21tu.
Hence the terms in this expression can be combined and the resulting expression is \(12t^{2}+21tu\)
Terms in Expression 3:
7fg ,8jk ,4lm ,4fg ,4jk – In this expression 7fg and 4fg can be combined to give 11fg.
8jk and 4jk can be combined to give 12jk.
The term 4lm will remain the same as there are no like terms.
Hence some of the terms in this expression can be combined and the resulting expression is 11fg +12jk +4lm.
Terms in Expression 4:
12rs , 3tx , 4gh , 3ty – None of these terms have any common variables.
Hence, the terms in this expression cannot be combined.
Example 3:
Simplify the expression:[(6x – 8) – 2x] – [(12x – 7) – (4x – 5)]
Solution:
The expression in the problem:
[(6x – 8)- 2x] – [(12x – 7) – (4x – 5)] [Write the expression]
[6x – 8 – 2x] – [12x – 7 – 1(4x) – 1(–5)] [Solving the innermost parentheses]
[6x – 2x – 8] – [12x – 7 – 4x + 5] [Solving by combining like terms]
[4x – 8] – [12x – 4x – 7 + 5] [Solving by combining like terms]
4x – 8 – [8x – 2] [Simplify]
4x – 8 – 8x + 2 [Simplify]
–4x – 6
Hence, the expression, [(6x – 8) – 2x] – [(12x – 7) – (4x – 5)] when simplified results in the expression –4x – 6.
Terms are defined as the parts of an algebraic expression and can be a combination of variables and coefficients, just variables or constants.
A constant is defined as a value that never changes in an expression. In the expression, 5x + 12 + 4xy, the number 12 is a constant as its value never changes in the expression.
All the four basic math operations, addition, subtraction, multiplication and division can be applied to an algebraic expression.
Combining like terms is a method used to simplify algebraic expressions. This method helps in making calculations easier by combining terms with the same variables.
For more information on algebra you can click on the following article links:
Basics of Algebra
Algebra
Algebraic Equations
Algebra-Symbols