About Basics of Algebra
Introduction
The word “Algebra” is derived from the Arabic word al-jabr, which loosely translates to “balancing and restoring”. We can think of algebra, like all math, as the mystery of the unknown. We find out these unknowns by doing things like addition, subtraction, multiplication, and division. In algebra, by learning methods of determining variables or unknowns, we develop the ability to solve many problems in daily life.
Variable
The word “variable” represents a value that can vary or change. A variable is generally an alphabet written in the lower case which represents an unknown value. Some examples of variables are:
a, b, c, x, y, z, and so on.
Let us understand this better by looking at this example.
Example: Represent the following situations using variables:
1. Some parents are attending a school concert, and they are seated in the school auditorium. There are 9 parents seated in a row. Using n for the number of rows, represent the total number of parents algebraically.
Solution:
As the number of rows is unknown to us, let us consider it to be a variable n. Since there are 9 parents seated in a row, the total number of parents is 9 x n or 9n.
2. If there are 30 peaches in a box, how will you write the total number of peaches in terms of the number of boxes? (Use p for the number of boxes.)
Solution:
As the number of boxes is unknown to us, let us consider it to be a variable p. Since there are 30 peaches in a box, the total number of boxes is 30 x p or 30p.
Constants
In algebra, a constant is a fixed value that remains unchanged. Sometimes, Greek letters or symbols are used to represent universal constants. Universal constants are constants whose values never change.
For example, pi is a constant and it is represented by the Greek letter “\(\pi\)”.
Algebraic Operations
In algebra, a basic algebraic operation includes addition, subtraction, division, and multiplication. To decide the order in which we prioritize these operations, we use PEMDAS (parenthesis, exponents, multiplication, division, addition, and subtraction).
Algebraic Expression
An expression that may contain numbers, constants, operands such as addition (+), subtraction (-), division (\(\div\)), and multiplication (\(\cdot\)), and one or more variables is called an algebraic expression.
The numerical factor of a term that contains a variable is called a coefficient, and a fixed term without any variable is called a constant.
Rapid Recall
Solved Examples
Example 1: Identify each part of the given algebraic expression 4x + 9.
Solution:
4x + 9
The above expression contains two terms, 4x and 9.
In this expression, the numerical factor is 4 which is multiplied by the variable x, so here the coefficient will be 4.
Similarly, 9 is a fixed value which cannot be changed, so it will be constant.
Hence,
Terms | 4x, 9 |
|---|---|
Coefficient | 4 |
Constant | 9 |
Variable | x |
Example 2: Evaluate m + 10 when m = 5
Solution:
The expression is m + 10
Substituting m = 5 in the above expression, we get,
m + 10 = 5 + 10
= 15
Hence, the value of m + 10 at m = 5 is 15.
Example 3: Ronald is saving up to buy a car for $140. He began with $20 and saved $5 each week. The expression 20 + 5k gives the amount of money he saves after k weeks. Can he buy the car after 25 weeks?
Solution:
An expression which represents Ronald’s savings after k weeks is 20 + 5k
To find the amount of money he saves after 25 weeks, we have to solve the expression when k = 25.
20 + 5k = 20 + \(5\times 25\)
= 20 + 125
= 145
Since the car costs $140, Ronald can buy it after 25 weeks.
The coefficient of a variable with a numerical factor of four is 4.
The word “variable” means the value that can vary or change. The value of the variable is not fixed. It is the unknown quantity or value in an algebraic expression, whereas a constant in algebra is a fixed value which remains unchanged.
The parts of the algebraic equations are terms, coefficients, variables, and constants.
In algebra, a basic algebraic operation includes addition, subtraction, division, and multiplication.
The steps for solving basic algebra are:
- Identify the variables and constants.
- Transform the variables on one side of the equation and the constants on the other side of the equation.
- Finally, apply arithmetic operations to simplify the equation.
- Get the answer.