Addition and subtraction of algebraic expressions (Definition, Examples) Byjus

Addition and subtraction of algebraic expressions

In mathematics, algebraic expressions involve variables, coefficients, and constants in association with mathematical operations. We can apply mathematical operations such as addition, subtraction, multiplication and division to algebraic expressions. In this article, we will learn how to add and subtract algebraic expressions....Read MoreRead Less

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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.

frc1

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.

frc2

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.

 

frc3

 

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.

Frequently Asked Questions

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.

Your child can practice and show you how well they understand a concept and what parts of the problem-solving process or operations they have learned. You can provide them with additional math worksheet practice in areas they are still developing.