Table of Contents
- Algebra
- Variables
- Constants
- Operations
- Algebraic expressions
- Coefficient of a variable
- Identifying parts of an algebraic expression
- Evaluating algebraic expressions
- Evaluating algebraic expressions with two variables
- Using two operations to evaluate algebraic expressions
- Examples:
- Frequently Asked Questions
Algebra
Algebra is one of the widest and ancient branches in the history of mathematics that includes a study of geometry, number theory, and analysis. It is generally defined as a study involving mathematical symbols and rules pertaining to the manipulation of these symbols.
An essential area of mathematics, algebra deals with a combination of variables, operations, and constants. Algebraic expressions also involve basic arithmetic operations such as addition, subtraction, multiplication, and division.
Variables
Variables are entities that represent an unknown quantity or object by means of the alphabet.
Example: Let’s suppose there are two animals: a dog and a cat. So variable “a” represents the dog and variable “b” represents the cat.
Constants
A number occurring in an algebraic expression that is “invariable or unchanging” is called a constant.
Examples of constant are, 9, 3, 1, – 2, – 5 etc.
Operations
Operators are mathematical symbols that emphasize a particular action on a set of constants or variables, such as addition, subtraction, division, multiplication. Operands on the other hand can be defined as a number or value upon which the mathematical operation will be applied.
In algebra, we use basic mathematical operations such as addition, subtraction, division, and multiplication using PEMDAS (parenthesis, exponents, multiplication, division, addition, subtraction), which is a predefined rule to solve various operations.
What is Algebraic expressions?
Definition: An algebraic expression is formed by using integers, constants, variables, and arithmetic operations. Variables and constants can be combined in an algebraic expression through mathematical operations.
For example, \(4\times a=4a\) is an algebraic expression.
Coefficient of a variable
The coefficient of a variable is the value of the integer that is multiplied with the variable. For example, if we consider an algebraic expression such as 4a, then 4 is the coefficient of “a” here.
Identifying parts of an algebraic expression
Identifying parts of an algebraic expression can be understood using expressions as in these examples:
Example 1: Identify the terms, coefficients, and constants in the following algebraic expression:
7a + 5b -21
The terms in the given expression are 7a, 5b, 21.
The coefficients are 7 and 5
The constant is 21.
Example 2: Through a pictorial representation, find the terms, coefficients, and constants in the following algebraic expression:\(2z^2+y+3\).
The image below represent the above algebraic expression:
The terms in the given expression are \(2z^2, y, 3\)
The coefficients are 2 and 1
The constant is 3.
Evaluating algebraic expressions
To evaluate an algebraic expression, we first substitute a number for each variable. Then, using the order of operations, we determine the value of the numerical expression.
Example: Evaluate the expression: a – 2, where a = 7
We have the expression a – 2. On substituting the value of “a” in the expression, we get the following answer:
7 – 2 = 5
Evaluating algebraic expressions with two variables
Evaluate the expression \(\frac{a}{b}\), where a = 49 and b = 7.
The expression is \(\frac{a}{b}\)
Given that a = 49 and b = 7,
\(\frac{a}{b}=\frac{49}{7}\)
\(\frac{a}{b}=7\)
Using two operations to evaluate algebraic expressions
Evaluate the expression\(\frac{5a}{b}\), where a = 5 and b = 5.
The expression is \(\frac{5a}{b}\)
Given that a = 5 and b = 5,
\(\frac{5a}{b}=\frac{5(5)}{b}\)
\(\frac{5a}{b}=\frac{25}{5}=5\)
Algebraic Expressions Examples
Example 1: To purchase a $35 watch, you save some money. You start with $5 and save $3 every week. The amount of money you save after “w” weeks is calculated by using the formula 5 + 3w. Will it be possible for you to purchase the watch after 10 weeks?
Solution:
Evaluate the expression: 5 + 3w, where w = 10, to find the amount of money you save after 10 weeks. Then, compare the expression’s value with the price of the watch.
5 + 3w = 5 + 3(10) Multiply 3 by 10
= 5 + 30 Simplify
= 35
Thus, it will be possible to buy the watch after 10 weeks because you will have the money that is needed to purchase the watch.
Example 2: The monthly membership cost of a club is calculated in dollars by using the formula 12 m and the registration charges are $30. Find the membership price for 8 months.
Solution:
Evaluate the expression, where m = 8 to find the amount you have to spend to buy a membership of 8 months. Then, compare the expression’s value with that of the membership.
12m + 30 = 12 (8) + 30 Add the registration charge.
= 96 + 30 Simplify.
= 126
Hence, the membership cost of the club will be $126 for a period of 8 months.
The coefficient of a single variable is 1.
We can represent the product in the following ways:
- 7 . n
- 7n
- 7(n)