Algebra – Definition with Examples

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What is Algebra?

The arithmetic operations of addition, subtraction, multiplication, and division help us solve mathematical problems. Algebra deals with these concepts and can be considered as generalized arithmetic.

algebra as generalized arithmetic

variable is an important concept of algebra. It can be an object or a letter that represents a number of things. We use variables to represent unknowns, to represent quantities that vary, and to generalize properties. The letters of the English alphabet, as well as Roman symbols, are commonly used for variables. 

An equation is a mathematical sentence with an equal sign. Example3 + 5 = 8

Inequality is a mathematical sentence that contains symbols <, >, ≤, ≥, or  ≠.Example4x + 7y ≥ 15

Equations and inequalities arise from everyday life situations. 

Example: Tina wants to buy pencils and pens for $15. Each pencil costs $1, and the pen costs $2 each. If x represents the number of pencils and y stands for pens, thenx + 2y ≤ 15 The algebraic thinking involved in correlating and translating the situations into equations and then solving them to find the solution for the everyday problems develops mathematical reasoning in children. Patterns are other tools that boost children’s mathematical thinking processes. A pattern is a regular arrangement of numbers, objects or shapes. There are mainly two types of patterns: 

Increasing or Decreasing Pattern:

12, 24, 36, 48,…

108, 102, 96, 90,…

increasing or decreasing pattern:

Repeating Pattern:

repeating pattern

Example 1: Mathew gets paid $2 per hour for his part-time work in a farm. If he works for 10 hours a week, how much money will he make? 

For an hour he gets $2, for two hours $4, for 3 hours $6, and so on.

Let us tabulate this sequence and find the earnings for the week.

No. of hoursPay ($)
12
24
36
48
510
612

Analyzing the pattern helps us find the amount that Mathew gets paid for any number of hours of work. The pattern in the sequence is that the pay is double the number of hours that he works. Therefore, for 10 hours of work, he will make $20.

Example 2: A library has two plans, 

Plan 1: You can register for $5 and rent any book for $2, 

Plan 2: Without registration, you can rent any book for $3. 

If Michelle is renting 7 books which plan would be beneficial?

The cost of 1 book in plan 1 is $7. Let us represent this as an ordered pair (1, $7) where the first number represents the number of books and the second number represents the cost. Then the cost for the first 7 books can be written as (1, $7), (2, $9), (3, $11), (4, $13), (5, $15), (6, $17), and (7, $19).  Similarly, for plan 2, the cost can be calculated as (1, $3), (2, $6), (3, $9), (4, $12), (5, $15), (6, $18), and (7, $21). Comparing the costs, plan 1 is beneficial.

Practice Problems

Algebra

Attend this Quiz & Test your knowledge.

1What will be the correct expression for the statement: Five more than twice of four.

5 + 2 + 4
5 + 2 ÷ 4
4 ÷ 2 + 5
2 × 4 + 5
CorrectIncorrect
Correct answer is: 2 × 4 + 5
Five more than twice of four can be written in expression as: 2 × 4 + 5 or 5 + 2 × 4

2Which expression correctly represents the situation: Sum of x and y divided by 2

x + y / 2
(x + y) / 2
x / 2 + y
2 / (x + y)
CorrectIncorrect
Correct answer is: (x + y) / 2
Sum of x and y is x + y.
Sum of x and y divided by 2 is (x + y) / 2.

3Which expression correctly represents the situation: a is divided by twice of b,

2a / b
2b / a
a / 2b
(a / 2) × b
CorrectIncorrect
Correct answer is: a / 2b
When a is divided by twice of b, it is a/2b.

4If x = 5 and y = 7, then x + y is ?

12
11
2
57
CorrectIncorrect
Correct answer is: 12
If x = 5 and y = 7, then x + y is ?

Frequently Asked Questions

Algebra is a branch of mathematics that deals with symbols or variables and uses arithmetic operations (+, –, ×, ÷) to find the unknown quantities represented by these variables. That is why algebra is also sometimes known as generalized arithmetic.

At this level of algebra, students learn to create, identify, and evaluate expressions. They also learn to identify and extend number patterns.

Algebra is very useful in real-life scenarios as it helps in representing problems or situations as mathematical expressions. These quantities could be the price of food items while grocery shopping or the profit margin for a businessman. So, unknown quantities like speed, time, distance, profit, money, and many others can all be calculated using equations in algebra.

In algebra, an expression is a combination of a minimum of two numbers or variables and at least one operator (+, –, ×, ÷). These expressions could either be arithmetic expressions or algebraic expressions. Arithmetic expressions are expressions that contain only numbers, for example, 3 + 8. In contrast, algebraic expressions are the ones that have both numbers and variables, for example, 3x + 8.


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