Algebra - Definition with Examples
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.
A 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.
An equation is a mathematical sentence with an equal sign.
Inequality is a mathematical sentence that contains symbols <, >, ≤, ≥, or ≠.
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, then
12, 24, 36, 48,...
108, 102, 96, 90,...
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 $6, and so on.
Let us tabulate this sequence and find the earning for the week.
No. of hours | Pay ($) |
1 | 2 |
2 | 4 |
3 | 6 |
4 | 8 |
5 | 10 |
6 | 12 |
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.
Algebra is a branch in 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.