What are the Basics of Algebra?

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

An algebraic expression is a collection of variable numbers and constants that combine with one, two, or more operations like addition (plus), subtraction (minus), multiplication, division, or a rational exponent. An algebraic expression includes a group of one, two, or more terms, where a term consists of factors — either number(s) (digits) or variable(s) or both together, with or without an exponent.

A variable is generally denoted by x or y or some other letter whose value is not known or can be anything.

A constant is a number by itself.

A coefficient is a number accompanying a variable (multiplication).

An exponent tells us how many times a variable or a number is multiplied by itself.

Examples

1. Consider the product of (2/3)² and (15/8)². Here, the fractional bases are different but the powers are the same. Thus, 

(2/3)² × (15/8)² = (2/3 × 15/8)² = (5/4)² = 5²/4² = 25/16.

2. Consider the product of (2/3)² and (2/3)⁵. Here, the fractional bases are the same. (2/3)² × (2/3)⁵ = (2/3)²⁺⁵. Thus,

(2/3)⁷ = 2⁷/3⁷ = 128/2187.

3. Consider the product of (3/4)² and (2/3)³. Here, the fractional bases and the powers are different. (3/4)² × (2/3)³  Thus, (3² × 2³)/(4² × 3³) = 1/6.

Interesting Fact

Egyptians were able to solve problems involving two equations with two unknowns, way back in 300 BC, but it was all done verbally without the use of any symbols, as they were not yet invented then.

How to Add

Addition in algebra is both related to and different from how basic addition is usually done with numbers. The difference is that only similar numbers or like terms can be added together, while the rest or unlike terms can only be included in the algebraic expression and cannot be combined.

Imagine a math worksheet word problem as follows:

A brother and a sister are going to a farmer’s market to buy their favorite fruits. Let’s assume that the brother bought 4 apples, 1 watermelon, and 3 oranges, while the sister got 2 apples, an orange, 5 strawberries, and 2 pears. Now when their mother asks them how much they got in all, it is a clear case of adding everything together.  

Since there are only two items (apples and oranges) that both the siblings purchased, only those two numbers can be added to each other. It goes without saying that apples and pears cannot be added to make an apple-pear, however tasty that may sound! The rest can only be grouped together, which leads us to a total of 6 apples (4+2), 4 oranges (3+1), 1 watermelon, 5 strawberries, and 2 pears.

This is how basic algebraic expressions are added together as well. We will have to find all the like terms in the expressions and sum the coefficients for each set of like terms, while the rest of the terms will be grouped together along with their relevant operators to get the answer.

Let us take two expressions:

x+5y+z and 2x-y+9w

In math, to add the two together, we need to find the like terms and add their coefficients first. The like terms here are (x and 2x) and (5y and -y) and the unlike terms are z and 9w. Please note that any variable by itself will be assumed to have a coefficient of 1, like “x” and “-y” from the given expressions. There are also a few laws of algebra that have to be followed while doing calculations in algebra, which have been listed separately.

While adding algebraic expressions we collect the like terms and add them. The sum of several like terms is the term whose coefficient is the sum of the coefficients of these like terms.

Two ways to practice math problems on the addition of algebraic expressions are:

Horizontal Method: In this method, all expressions are written in a horizontal line, and then the terms are regrouped to collect all the groups of like terms and then added.

Column Method: In this method, each expression is written in a separate row such that they are regrouped one below the other in a column. Then the addition of numbers is done column-wise. 

Examples of Addition of Algebraic Expressions:

1. Add 6a + 8b - 7c, 2b + c - 4a and a - 3b - 2c

Horizontal Sum Method

Solution: 

(6a + 8b - 7c) + (2b + c - 4a) + (a - 3b - 2c) 
= 6a + 8b - 7c + 2b + c - 4a + a - 3b - 2c

Arrange the like terms together, then add. 

Thus, the required addition is:

6a - 4a + a + 8b + 2b - 3b - 7c + c - 2c
= 3a + 7b - 8c

Column Sum Method:
Writing the terms of the given expressions in the same order in form of rows with like terms below each other and adding column-wise;

    6a + 8b - 7c
 -  4a + 2b +  c
      a  - 3b  - 2c
    3a + 7b - 8c
  The sum is 3a + 7b - 8c.

2. Add 5x² + 7y - 6z², 4y + 3x², 9x² + 2z² - 9y and 2y - 2x²

Horizontal Method

Solution:

(5x² + 7y - 6z²) + (4y + 3x²) + (9x² + 2z² - 9y) + (2y - 2x²).
= 5x² + 7y - 6z² + 4y + 3x² + 9x² + 2z² - 9y + 2y - 2x²

Arrange the like terms together, then add.

Thus, the required addition is:

5x² + 3x² + 9x² - 2x² + 7y + 4y - 9y + 2y - 6z² + 2z²
= 15x² + 4y - 4z²

Column Method

Solution:

Arrange expressions in lines so that the like terms with their signs are one below the other i.e. like terms are in the same vertical column and then add the different groups of like terms.

   5x² + 7y - 6z²
+ 3x² + 4y
+ 9x² -  9y + 2z²
- 2x² +  2y
________________
15x² +  4y - 4z²

Rules of Algebra

There are a few rules that have to be followed while doing operations on algebraic expressions. These are:

• Commutative law of addition
• Commutative law of multiplication
• Associative law of addition
• Associative law of multiplication
• Distributive laws

For the purpose of this article, i.e., the addition of algebraic expressions, only the two laws of addition will come into play, namely the commutative and associative laws of addition. Assuming a, b & c are variables, the laws will be applicable in the following way:

Commutative Law of Addition

a + b  =  b + a

The arrangement of the terms does not affect the sum of the numbers. In other words, even if the order of the terms were changed, there would be no difference in the answer.

Associative Law of Addition

(a + b) + c  =  a + (b + c)

The grouping of the terms does not affect the addition or the sum. In other words, the parentheses can shift places, with the terms remaining where they are, to get the same number.

Applications in Real-Time

Algebra, as in math, is used knowingly or unknowingly in everyday life for a whole gamut of things. One of the most basic uses of the algebraic expression in daily life is in the form of setting up a daily schedule or routine.

Suppose your math class starts at a particular time every Monday. There are a whole lot of variables involved in you reaching the class on time, starting from how late you sleep to the timing and efficiency of the bus that will take you to school. The time taken by you to finish your morning activities will of course play an important role in whether you reach your class on time.

Since the class time is constant or not subject to change, the other things can be varied and adjusted to make sure you get to attend the class, like skipping your breakfast instead of waking up early. Only do so at the risk of getting on the wrong side of your mother, though.

Common Mistakes

• One of the most common mistakes made by math students during the addition of algebraic expressions is incorrect grouping of like terms, or grouping together of unlike terms.
• Double-check all the variables and their exponents carefully before adding the like terms.
• The use of parentheses helps greatly in avoiding calculation mistakes in algebra.
• PEMDAS order is followed in algebraic addition—Parentheses, Exponents, Multiplication or Division, Addition or Subtraction. This order of precedence for operations has to be kept in mind just like with numerical calculations.

Context and Applications

The concept of algebraic addition is greatly used in K-12 examinations, as well as in graduate and postgraduate studies, especially for:

• B.A Mathematics
• B.Sc. Mathematics
• Masters in Mathematics

Related Concepts

• Mathematical theories, methods, tools, and practices
• Numeracy and numerical concepts
• Data analysis and interpretation
• Linear Algebra