Subtraction of Algebraic Expressions

Subtraction of Algebraic Expressions

Jamie recently learned about algebraic expressions and is now wondering if he can subtract those expressions in the same way he subtracts two whole numbers or fractions.

Can we subtract two algebraic expressions?

Do you want to help Jamie in subtracting any two or more algebraic expressions?

Let's find out how to subtract algebraic expressions.

In this mini-lesson, we will explore subtraction of algebraic expressions for grade 7 by learning about like and unlike terms, methods to subtract algebraic expressions with the help of interesting simulation, some solved examples and a few interactive questions for you to test your understanding.

Lesson Plan

What Do You Mean by Like and Unlike Terms?

Before learning about like and unlike terms in an algebraic expression, let us first briefly understand the meaning of algebraic expression.

An algebraic expression (or) a variable expression is a combination of terms by operations such as addition, subtraction, multiplication, division, etc.

An example of an algebraic expression is \(5x+7\).

Example of an algebraic expression

We can categorize the terms in an algebraic expression into two types - like and unlike terms.

Terms containing same variable raised to the same power are known as like terms.  

In like terms, one can only change the numerical coefficient

Examples of like terms are

\(5x\) and \(13x\)

\(7y^3\) and \(3y^3\)

\(a^4\) and \(6a^4\)

in the above examples, only the numerical coefficient differs and we have same variable in each pair of like terms raised to the same power.

Terms that have different variable or same variable raised to different powers are known as unlike terms.

Examples of unlike terms are

\(5x\) and \(5y\)

\(2m^5\) and \(8m^3\)

\(56p^3\) and \(q^3\)

Let us go back to our first example, \(5x+7\).

This expression has 2 terms, \(5x\) and \(7\). 

Are these like terms or unlike terms?

\(5x\) and \(7\) are unlike terms as they do not have a common variable. First term \(5x\) has variable \(x\), while second term \(7\) does not have any variable. So these are unlike terms.

Like and unlike terms


How Do You Subtract Algebraic Expressions?

Can we subtract \(3\) apples from \(4\) bananas?

Subtract 3 apples from 4 bananas   

The answer is NO.

We cannot subtract \(3\) apples from \(4\) bananas, as they are two different objects.

Similarly, in case of terms in an algebraic expression, we cannot subtract two or more unlike terms.

An important point to remember while subtracting algebraic expressions is that we can only subtract like terms.

There are two methods to do subtraction of algebraic expressions - horizontal method and column method.

Horizontal Method

Steps to be followed while doing algebraic expression subtraction by horizontal method is written below:

Step 1- Write all the expressions in a horizontal line by putting them into brackets and put subtraction sign in between.
Step 2- Open the brackets. Change the operators whenever there is a negative sign outside the bracket, for example, \(+\) to \(-\) and \(-\) to \(+\).
Step 3- Group all the like terms together from all the expressions and rewrite it in a single expression.
Step 4- Add numerical coefficients of all the like terms followed by the common variable.
Step 5- Rewrite the simplified expression, and make sure all the terms in the final answer are unlike terms.

Column Method

Steps to be followed to do subtraction of algebraic expressions by column method is written below:

Step 1- Write both expressions one below the other. Make sure you have written like terms in one column.

\(\begin{align}2x^2+3x-4y+7 \end{align}\)

           \(\begin{align}5x+4y-3 \end{align}\)

            If there is a term whose like term is not there in the second expression, for example, \(2x^2\), then either write \(0\) below it or leave that column blank.

Step 2- Change the operators in the last row (second expression), for example, \(+\) to \(-\) and \(-\) to \(+\).

\(\begin{align}2x^2+3x-4y+7 \end{align}\)

         \(\begin{align} \pm 5x\pm 4y\mp 3 \end{align}\)

Step 3- Consider the changed signs and add the numerical coefficient of each column (like terms) and write below it in the same column followed by the common variable.

\(\begin{align}2x^2+3x-4y+7 \end{align}\)

\(\begin{align} \underline{\ \ \ \ \ \ \ \pm 5x\pm 4y\mp 3} \end{align}\)

\(\begin{align} 2x^2-2x-8y+10 \end{align}\) 

Step 4- Rewrite the final answer. Ignore writing \(0\) as a term in the final answer, if any.

Subtraction of Algebraic Expressions Calculator

Try your hands at this simulation of a calculator for subtracting algebraic expressions. Write both the expressions that you want to subtract and click on "Subtract" to get your answer.

 
important notes to remember
Important Notes
  1. Subtraction of algebraic expressions can be done by two methods: horizontal method and column method.
  2. It is always better to subtract two expressions at one time. Never try to subtract three or more expressions together through column method.
  3. Operators inside the brackets need to be changed if there is a negative sign outside the brackets.
  4. If there is no sign written with the first term of the algebraic expression, we consider it as positive. For example \(3x\) is same as \(+3x\).

Solved Examples 

Example 1

 

 

Find the length of side \(AC\) of the triangle whose perimeter is \(4x^2+17xy+5\) units.

Subtraction of algebraic expression example

Solution

Perimeter of the triangle=\(4x^2+17xy+5\) units

We know that perimeter of a triangle is the sum of all its sides. So, to find the length of side \(AC\), we need to add the length of the other two sides and subtract that from the perimeter.

Sum of the other two sides can be calculated as,

\((x^2+5xy-2)+(5xy-3x^2)\)

\(x^2-3x^2+5xy+5xy-2\)

\(-2x^2+10xy-2\)

Now, we subtract it from the perimeter of the triangle,

\((4x^2+17xy+5)-(-2x^2+10xy-2)\)

\(4x^2+17xy+5+2x^2-10xy+2\)

\(4x^2+2x^2+17xy-10xy+5+2\)

\(6x^2+7xy+7\)

\(\therefore\) The length of side \(AC\) is \(6x^2+7xy+7\) units.
Example 2

 

 

A metal rod of length \(5mn-2n+1\) units is cut into two parts. If the length of the bigger part is \(3mn+n\) units, find the length of the smaller part of the rod. 

Length of the smaller part of the rod

Solution

To find the length of the smaller part of the rod, we need to subtract the length of the larger part from the total length of the rod.

Subtraction by column method

\(\therefore\) The length of the smaller part of the rod is \(2mn-3n+1\) units.
Example 3

 

 

Ron is taller than his brother. One day, they measured their heights on a scale. Ron's height is \(4x^3+y^3\) units and his brother's height is \(x^3+3y^3\) units. Find the difference between the heights of Ron and his brother. 

Example of subtraction of algebraic expressions

Solution

To find the difference in the height of both children, we need to subtract the height of the shorter boy from the height of the taller boy.

\((4x^3+y^3)-(x^3+3y^3)\)

\(4x^3-x^3+y^3-3y^3\)

\(3x^3-2y^3\)

\(\therefore\) The difference between their heights is \(3x^3-2y^3\).
 
Challenge your math skills
Challenging Questions
  1. Subtract \((\frac{5}{7}x^3-\frac{1}{2}xy)-(\frac{1}{2}x^3+3xy)\).
  2. Perimeter of a square is \((7mn-4np+3) \text{units}\). If the sum of three sides is given as \((8mn+3-2np) \text{units}\), find the length of fourth side of the square.

Now, try to find answers of questions given below taken from subtraction of algebraic expressions worksheets designed by our experts for your practice.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of subtraction of algebraic expressions. The math journey around subtraction of algebraic expressions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions(FAQs)

1. What are the four basic rules of algebra?

The four basic rules or properties of algebra are commutative law, associative law, distributive law, and identity rule.

2. How do you simplify an expression?

To simply an expression, we group together all like terms such that the simplified expression will only have unlike terms in it.

3. What are the types of algebraic expressions?

Based on the degree of polynomial, algebraic expressions can be classified as linear expressions, quadratic expressions, and cubic expressions. While, on the basis of terms, it can be classified as monomial expression, binomial expression, and trinomial expression.

4. What is the rule for adding and subtracting algebraic terms?

The basic rule to add and subtract algebraic terms is to add only like terms. Also, if there is a negative sign outside the bracket (in case of subtraction), we change the operators of the terms inside the bracket. 

5. Do you add or subtract like terms?

Yes, we can add or subtract like terms by adding/subtracting their numerical coefficients.

6. How do you divide algebraic expressions?

Division of algebraic expressions is done by taking out the common factors from both numerator and denominator by factorization.

7. How do you add and subtract algebraic expressions with unlike denominators?

To add and subtract algebraic expressions with unlike denominators, we first make sure that the denominators of like terms are same, i.e., the LCM of the denominators. We then add or subtract numerical coefficients of like terms.

8. How do you simplify like terms?

To simplify like terms, we perform the given operation on the numerical coefficient of the terms. For example, \(3m^2+2m^2=5m^2\).

9. How do you find unlike terms?

Terms that have different variable or same variable raised to different powers are known as unlike terms. For example, \(8ab\) and \(8a\).

10. Are 5x and 4xy like terms?

No, \(5x\) and \(4xy\) are not like terms since they have different variables, \(x\) and \(xy\) respectively.

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