Subtraction of Algebraic Expressions(With Methods & Examples)

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In mathematics, addition, subtraction, multiplication, and division are four basic operations. Just like we subtract numbers, we can perform the subtraction of algebraic expressions.  In order to subtract two algebraic expressions, we combine the like terms and then subtract them.

Let’s understand the methods of subtracting algebraic expressions with steps and examples.

What is the Subtraction of Algebraic Expressions?

The subtraction of algebraic expressions is quite similar to the subtraction of numbers. However, while subtracting algebraic expressions, you need to collect the like terms and then subtract them. The difference between like terms would be the like term whose coefficient is the difference between the coefficients of the like terms and variables same as that of subtrahends.

Can we subtract $2$ pens from $5$ notebooks? The answer is NO. We cannot subtract $2$ pens from $5$ notebooks, as they are two different objects. Similarly, in the case of terms in an algebraic expression, we cannot subtract two unlike terms.

There are two ways for performing algebra subtraction.

• Horizontal Method of Algebra Subtraction
• Column Method of Algebra Subtraction

Horizontal Method of Algebra Subtraction

In this method, we write all expressions in a horizontal line and then arrange the terms to collect all the groups of like terms. These like terms are then subtracted.

Steps of Horizontal Method of Algebra Subtraction

These are the steps to perform the subtraction of two algebraic expressions using the horizontal method.

Let’s consider two algebraic expressions $7x^{4} – 2x^{3} + 5x + 9$ and $2x^{4} + 6x^{2} + 9x – 3$ and we want to subtract $2x^{4} + 6x^{2} + 9x – 3$ from $7x^{4} – 2x^{3} + 5x + 9$, i.e., we want to perform the operation $\left(7x^{4} – 2x^{3} + 5x + 9 \right) – \left(2x^{4} + 6x^{2} + 9x – 3 \right)$

Step 1: Write the given algebraic expressions using a subtraction symbol.

$\left(7x^{4} – 2x^{3} + 5x + 9 \right) – \left(2x^{4} + 6x^{2} + 9x – 3 \right)$

Step 2: Open the brackets and multiply the signs(Use the rules for opening the brackets).

$7x^{4} – 2x^{3} + 5x + 9 – 2x^{4} – 6x^{2} – 9x + 3 $

Step 3: Now, combine the like terms.

$\left(7x^{4} – 2x^{4} \right) – 2x^{3} – 6x^{2} + \left(5x – 9x \right) + \left(9 + 3 \right) $

Step 4: Add the coefficients. Keep the variables and exponents on the variables the same.

$5x^{4} – 2x^{3} – 6x^{2} – 4x + 12$

Step 5: Rewrite the answer by arranging the terms in descending order of exponents.

$5x^{4} – 2x^{3} – 6x^{2} – 4x + 12$

Therefore,  $\left(7x^{4} – 2x^{3} + 5x + 9 \right) – \left(2x^{4} + 6x^{2} + 9x – 3 \right) = 5x^{4} – 2x^{3} – 6x^{2} – 4x + 12$.

Examples

Ex 1: Subtract $3s^{2} – 4s + 8$ from $s^{2} – 5s + 13$.

$\left(s^{2} – 5s + 13 \right) – \left(3s^{2} – 4s + 8 \right)$

$=s^{2} – 5s + 13 – 3s^{2} + 4s – 8$

$=\left(s^{2} – 3s^{2} \right) + \left(- 5s + 4s \right) + \left(13 – 8 \right)$

$=- 2s^{2} – 1s + 5 =- 2s^{2} – s + 5$

Ex 2: Subtract $x^{4} – 5x^{3}y + 2y^{3} – 6x^{2} + 7xy + 8$ from $8x^{4} + 8x^{3} + 6y^{3} + 9xy + 12$

$\left(8x^{4} + 8x^{3} + 6y^{3} + 9xy + 12 \right) – \left(x^{4} – 5x^{3}y + 2y^{3} – 6x^{2} + 7xy + 8 \right)$

$= 8x^{4} + 8x^{3} + 6y^{3} + 9xy + 12 – x^{4} + 5x^{3}y – 2y^{3} + 6x^{2} – 7xy – 8$

$= \left(8x^{4} – x^{4} \right) + 8x^{3} + \left(6y^{3} – 2y^{3} \right) + \left(9xy – 7xy \right) + \left(12 – 8 \right) + 5x^{3}y + 6x^{2}$

$= 7x^{4} + 8x^{3} + 4y^{3} + 2xy + 4 + 5x^{3}y + 6x^{2}$

$= 7x^{4} + 5x^{3}y + 8x^{3} + 4y^{3} + 6x^{2} + 2xy + 4$

Ex 3: Subtract $\left(13.8x^{2} + 6.2x + 5 \right)$ from $\left(10.9x^{2} – 5x – 3 \right)$

$\left(10.9x^{2} – 5x – 3 \right) – \left(13.8x^{2} + 6.2x + 5 \right)$

$= 10.9x^{2} – 5x – 3 – 13.8x^{2} – 6.2x – 5$

$= \left(10.9x^{2} – 13.8x^{2} \right) + \left(- 5x – 6.2x \right) + \left(- 3 – 5 \right)$

$= -2.9x^{2} – 11.2x – 8$

Column Method of Algebra Subtraction

In this method, we write each expression in a separate row in a way that their like terms are arranged one below the other in the column. Then you need to subtract the terms column-wise.

Steps of Column Method of Algebra Subtraction

These are the steps to perform the subtraction of two algebraic expressions using the column method.

Let’s consider two algebraic expressions $5a^{4} + 6a^{3}b^{2} – 8b^{3} + 9a^{2}b^{2} + 12$ and $7a^{4} + 2a^{3}b^{2} + 4a^{3} – 2a^{2}b^{2} + 18$ and we want to subtract $5a^{4} + 6a^{3}b^{2} – 8b^{3} + 9a^{2}b^{2} + 12$ from $7a^{4} + 2a^{3}b^{2} + 4a^{3} – 2a^{2}b^{2} + 18$, i.e., $\left(7a^{4} + 2a^{3}b^{2} + 4a^{3} – 2a^{2}b^{2} + 18 \right) – \left(5a^{4} + 6a^{3}b^{2} – 8b^{3} + 9a^{2}b^{2} + 12 \right)$

Step 1: Write all the expressions one below the other. Make sure to like terms in one column. If there is a term whose like term is not there in the second expression, then leave that column blank.

Step 2: Subtract the numerical coefficient of each column (like terms) and write below it in the same column followed by the common variable. 

Step 3: Rewrite the answer by arranging the terms in descending order of exponents.

Therefore, the answer is $2a^{4} – 4a^{3}b^{2} + 4a^{3} + 8b^{3} – 11a^{2}b^{2} + 6$.

Rewriting the answer by arranging the terms in descending order of exponents

$- 4a^{3}b^{2} + 2a^{4} – 11a^{2}b^{2} + 4a^{3} + 8b^{3} + 6$.

Therefore, the answer is $2a^{4} – 4a^{3}b^{2} + 4a^{3} + 8b^{3} – 11a^{2}b^{2} + 6$.

Rewriting the answer by arranging the terms in descending order of exponents

$- 4a^{3}b^{2} + 2a^{4} – 11a^{2}b^{2} + 4a^{3} + 8b^{3} + 6$.

Examples

Ex 1: Subtract $\left(5x^{2} + 2x – 9 \right)$ from $\left(13x^{2} – 4x + 8 \right)$

Writing $\left(13x^{2} – 4x + 8 \right)$ and $\left(5x^{2} + 2x – 9 \right)$ one below the other.

Therefore, $\left(13x^{2} – 4x + 8 \right) – \left(5x^{2} + 2x – 9 \right) = 8x^{2} – 6x + 17$.

Ex 2: Subtract $\left(2m^{3} + 3m^{2}n + 2mn – 1 \right)$ from $\left(9m^{3} + 2mn^{2} + mn – 1 \right)$

Writing $\left(9m^{3} + 2mn^{2} + mn – 1 \right)$ and $\left(2m^{3} + 3m^{2}n + 2mn – 1 \right)$ one below the other.

Therefore, $\left(9m^{3} + 2mn^{2} + mn – 1 \right) – \left(2m^{3} + 3m^{2}n + 2mn – 1 \right) = 7m^{3} + 2mn^{2} – 3m^{2}n – mn$.

Tips for Subtraction of Algebraic Expressions

• We can ignore the order of variables in like terms in an algebraic expression. For example,  $3a + 2b$, and, $9b + a$ both are like terms.
• We can ignore writing $1$ as the numerical coefficient of any term. For example, $xy$ is the same as $1xy$.
• We can replace a missing term with $0$ with the same variables. For example, a missing term can be written as $0x$, $0y$, or $0xy$ depending on the variables of the missing term.

Practice Problems

1. Subtract the following algebraic expression using the horizontal method.
   • $2x^{2} + 3xy + 5y^{2}$ from $-4x^{2} + xy + 8y^{2}$
   • $x^{2} + 7xy – 2y^{2}$ from $6x^{2} + 4xy – 2y^{2}$
   • $-5x^{2} – 6y^{2}$ from $9xy – 12y^{2}$
   • $9x^{2} – xy + 5y^{2}$ from $12x^{2} + 2xy $
2. Subtract the following algebraic expression using the column method.
   • $6x^{2} + 13xy + 12y^{2}$ from $-x^{2} + 5xy – 7y^{2}$
   • $x^{2} – y^{2}$ from $2xy + 7y^{2}$
   • $10xy – y^{2}$ from $15x^{2} + 10xy + 8y^{2}$
   • $3x^{2} + 15xy – 8y^{2}$ from $4x^{2} + 8xy $

FAQs

Conclusion

As you subtract and simplify numbers, algebraic expressions can also be subtracted and simplified. To subtract two algebraic expressions, we combine all the like terms and then subtract them and then arrange the terms in descending order of the exponents of the variables.

Recommended Reading

• Addition of Algebraic Expressions(With Methods & Examples)
• What is Algebraic Expression(Definition, Formulas & Examples)
• What is Algebra – Definition, Basics & Examples
• What is Pattern in Math (Definition, Types & Examples)
• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)