Algebra is the branch of mathematics in which letters and symbols are used to represent numbers and quantities in equations and expressions. It allows us to write general rules and solve unknown values.
Variables
A variable is a symbol (usually a letter like x, y, n) that represents an unknown or changing quantity. For example, if a person has n pencils and gets 3 more, the total is n + 3.
Constants
A constant is a fixed value that does not change. In 5x + 3, the 3 is a constant.
- Algebraic Expressions
- An algebraic expression is a combination of constants and variables using arithmetic operations.
- 3x + 5 is an expression with variable x.
- 2a - b + 7 is an expression with two variables.
Terms
A term is a single number, variable, or product of numbers and variables. In 4x2 + 3x - 7:
- 4x2, 3x, and -7 are the three terms.
Equations
An equation is a statement that two expressions are equal, using "=".
Example: 2x + 5 = 13.
A solution is the value of the variable that makes the equation true. Here, x = 4 (since 2x4+5=13).
Simple Linear Equations
To solve: perform the same operation on both sides to isolate the variable.
If 3x = 18, divide both sides by 3: x = 6.
If x + 7 = 15, subtract 7 from both sides: x = 8.
Write an algebraic expression: "7 more than 3 times a number n."
3n + 7.
Find the value of 4x - 3 when x = 5.
4(5) - 3 = 20 - 3 = 17.
Solve: x + 12 = 20.
x = 20 - 12 = 8.
Solve: 5y = 35.
y = 35/5 = 7.
A number is multiplied by 4 and then 6 is subtracted. The result is 22. Find the number.
Let the number be n. 4n - 6 = 22. 4n = 28. n = 7.
The perimeter of an equilateral triangle is 3s. If the perimeter is 45 cm, find s.
3s = 45. s = 15 cm.
Riya is x years old. Her mother is 3 times her age. If the sum of their ages is 48, find Riya's age.
x + 3x = 48. 4x = 48. x = 12. Riya is 12 years old.
Patterns and Algebra
Algebraic expressions can describe patterns. The nth term of 3, 6, 9, 12, ... is 3n.
- Key Rules
- Like terms (same variable, same power) can be combined: 3x + 5x = 8x.
- Unlike terms cannot be combined: 3x + 5y remains as 3x + 5y.
Common mistakes
- x + x = 2x (not x2).
- 3x means 3 times x, not 3 added to x.
- When solving equations, always perform the same operation on BOTH sides.
Summary
Algebra uses variables (letters) to represent unknowns. Expressions combine variables and constants using operations. Equations set expressions equal. Solving an equation means finding the value of the variable. Algebraic thinking helps describe patterns and solve real-world problems.