What are Algebraic Expressions?
As mentioned earlier, all algebraic expressions comprise variables, coefficients and constants related to each other by addition, subtraction, multiplication and division operations. Expressions also have a finite number of terms. We can add, subtract, multiply or divide two algebraic expressions.
There are two types of algebraic expressions, linear expressions and nonlinear expressions based on the exponent of the variables in an expression.
- Linear Algebraic Expressions
If the exponent of the variable in an algebraic expression is equal to 1 then it is called a linear algebraic expression. For example, 2x + 3 or x + 3y + 1.
- Nonlinear Algebraic Expressions
If the algebraic expression has one or more variables with an exponent of 2 or more, then it is referred to as a nonlinear algebraic expression.
For example, \(1+x^{2}\) and \(y^{3}-3\)
Addition and Subtraction of Algebraic Expressions
We can add or subtract algebraic expressions using two different methods.
- Vertical Method
In this method we will first align the like terms vertically and then add or subtract as applicable.
For example: Find the solution for \(\left ( x-6 \right )+\left ( 5x+7 \right )\)
Solution:Align the like terms vertically and add.
Hence, the sum is 6x+1.
- Horizontal Method
In this method we use the properties of mathematical operations to group the like terms, simplify them, and then add or subtract.
For example: Find the solution for \(\left ( x-6 \right )-\left ( 5x+7 \right )\)
Solution: Use the properties of operations to group the like terms and simplify.
\(\left ( x-6 \right )-\left ( 5x+7 \right )\) [Write the expression]
\(=\left ( x-6 \right )+\left ( 5x+7 \right )\) [Add the opposite]
\(=x+\left ( -5 \right )x-6+\left ( -7 \right )\) [Commutative property of addition]
\(=\left [ x+\left ( -5 \right )x \right ]+\left [ -6+\left ( -7 \right ) \right ]\) [Group the like terms]
\(=-4x-13\) [Combine like terms]
The difference is -4x-13.
Solved Examples
Example 1: Find the sum of 4x-9 and 5x-3.
Solution:
Vertical Method
Align the like terms vertically and add.
So, the sum is 9x-12.
Example 2: Find the difference, \(\left ( 17x-6 \right )-\left ( 11x-3 \right )\)
Solution:
Horizontal method
Use the properties of operations to group the like terms and simplify.
\(\left ( 17x-6 \right )-\left ( 11x-3 \right )\) [Write the expression]
\(=17x-6+\left ( -11x+3 \right )\) [Add the opposite]
\(=17x+\left ( -11 \right )x-6+3\) [Commutative Property of Addition]
\(=17x+\left ( -11 \right )x+\left ( -6+3 \right )\) [Group the like terms]
\(=6x-3\) [Combine like terms]
So, the difference is 6x-3.
Example 3: William draws a triangle with perimeter \(=\left ( 17x+13 \right )\) centimeters. The sides of the triangle are \(=\left ( 3x+9 \right )\) centimeters and \(=\left ( 2x-3 \right )\) centimeters. Find the length of the base of this triangle.
Solution:
Let the known side of the triangle be ‘y’ cm
\(17x+13=\left ( 3x+9 \right )+\left ( 2x-3 \right )+y\) [Formula for perimeter of a triangle]
\(17x+13=3x+9+2x-3+y\) [Rewrite the sum]
\(17x+13=3x+2x+9-3+y\) [Commutative Property of Addition]
\(17x+13\left [ 3x+2x \right ]+\left [ 9-3 \right ]+y\) [Group like terms]
\(17x+13=5x+6+y\) [Add like terms]
\(17x+13-\left ( 5x+6 \right )=y\) [Solve for y]
\(17x+13+\left ( -5x-6 \right )=y\) [Add the opposite]
\(17x-5x+13-6=y\) [Commutative Property of Addition]
\(\left [ 17x-5x \right ]+\left [ 13-6 \right ]=y\) [Group like terms]
\(12x+7=y\) [Combine like terms]
\(y=12x+7\) [Rewrite the equation]
So, the measure of the unknown side of the triangle is \(\left ( 12x+7 \right )\) centimeters.
A variable is a letter usually from the English alphabet that represents an unknown quantity. Different values associated with the variable will provide different results.
An algebraic expression is a mathematical expression that consists of variables, coefficients and constants, along with mathematical operations such as addition, multiplication, subtraction, and division.
A constant is a number with a value that never changes in an expression.