Addition of Algebraic Expressions(With Methods & Examples)

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

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

What is the Addition of Algebraic Expressions?

The addition of algebraic expressions is quite similar to the addition of numbers. However while adding algebraic expressions, you need to collect the like terms and then add them. The sum of the several like terms would be the like term whose coefficient is the total of the coefficients of the like terms and variables same as that of addends.

There are two ways for performing the algebra addition.

• Horizontal Method of Algebra Addition
• Column Method of Algebra Addition

Horizontal Method of Algebra Addition

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 added.

Steps of Horizontal Method of Algebra Addition

These are the steps to perform the addition of two or more algebraic expressions using the horizontal method.

Let’s consider three algebraic expressions $2x^{3} + 3x^{2}y – 5xy{2} + 7xy + 6y – 9$, $3x^{2} + xy – 8y^{2}$, and $x^{3} + y^{3}$.

Step 1: Write the given algebraic expressions using an addition symbol.

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

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

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

Step 3: Now, combine the like terms.

$(2x^{3} + x^{3}) + 3x^{2}y – 5xy{2} + (7xy + xy) + 6y – 9 + 3x^{2} – 8y^{2} + y^{3}$

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

$3x^{3} + 3x^{2}y – 5xy{2} + 8xy + 6y – 9 + 3x^{2} – 8y^{2} + y^{3}$

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

$3x^{3} + y^{3} + 3x^{2}y – 5xy{2} + 3x^{2} – 8y^{2} + 8xy + 6y – 9$

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

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

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

$7x^{3} + y^{3} + 3x^{2}y – 16xy^{2} + 11$

Examples

Ex 1: Add $mn + t$, $2mn – 2t$ and $-3t + 3mn$

$\left(mn + t \right) + \left(2mn – 2t \right) + \left(-3t + 3mn \right)$

$=mn + t + 2mn – 2t – 3t + 3mn$

$=\left(mn + 2mn + 3mn \right) + \left(t – 2t – 3t \right)$

$=6mn – 4t$

Ex 2: Add $\left(-5x^{2} – x + 4 \right)$ and $\left(-3x^{2} – 5x + 2 \right)$

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

$=-5x^{2} – x + 4 – 3x^{2} – 5x + 2$

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

$=-8x^{2} – 6x + 6$

Ex 3: Add $\left(20.2x^{2} + 6x + 5 \right)$ and $\left(1.7x^{2} – 3x – 8 \right)$

$=\left(20.2x^{2} + 6x + 5 \right)$ + $\left(1.7x^{2} – 3x – 8 \right)$

$=20.2x^{2} + 6x + 5 + 1.7x^{2} – 3x – 8$

$=\left(20.2x^{2} + 1.7x^{2} \right) + \left(6x – 3x \right) + \left(5 – 8 \right)$

$=21.9x^{2} + 3x + (-3)$

$=21.9x^{2} + 3x – 3$

Column Method of Algebra Addition

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 add the terms column-wise.

Steps of Column Method of Algebra Addition

These are the steps to perform the addition of two or more algebraic expressions using the column method.

Let’s consider three algebraic expressions $5x^{3} + 2x^{2}y – 7xy^{2} + 8$, $-2x^{3} – 9xy^{2} + 6$,  $3x^{3} + x^{2}y – 3$, and $x^{3} + y^{3}$

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: Add 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 $7x^{3} + 3x^{2}y – 16xy^{2} + 11 + y^{3}$.

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

$7x^{3} + y^{3} + 3x^{2}y – 16xy^{2} + 11$.

Examples

Ex 1: Add $\left(-x^{2} + x – 4 \right)$ and $\left(3x^{2} – 8x – 2 \right)$

Writing $\left(-x^{2} + x – 4 \right)$ and $\left(3x^{2} – 8x – 2 \right)$ one below the other.

Therefore, $\left(-x^{2} + x – 4 \right) + \left(3x^{2} – 8x – 2 \right) = 2x^{2} – 7x – 6$

Ex 2: Add $\left(6m^{5} + 1 \right)$ and $\left(2m^{5} + 9m – 1 \right)$

Writing $\left(6m^{5} + 1 \right)$ and $\left(2m^{5} + 9m – 1 \right)$ one below the other.

Therefore, $\left(6m^{5} + 1 \right) + \left(2m^{5} + 9m – 1 \right) = 8m^{5} + 9m$

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

FAQs

What is the rule for adding algebraic terms?

The basic rule to add algebraic terms is to add only like terms.

Can we add the unlike terms of the algebraic expressions?

No, we cannot add the unlike terms of the algebraic expressions. For example, $2x^{2} + y^{3}$ cannot be simplified further.

How do you combine the like terms and simplify?

Group together all the like terms, add or subtract the numerical coefficients of the like terms and attach the common variable to it.

Conclusion

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