Variables and Expressions in Mathematics

1. Understanding Variables

A variable is a symbol (usually a letter) that represents an unknown value or a value that can change. Variables allow us to express mathematical relationships and solve problems.

Key Concept: In mathematics, variables act as placeholders for numbers. They allow us to write general rules and formulas.

Common Uses of Variables:

Unknown values in equations
Representing changing quantities
Writing formulas and functions
Expressing patterns and relationships

Examples of Variables:

1. In the equation x + 5 = 12, x is a variable representing the unknown value (which is 7).

2. In the formula A = πr², r is a variable representing the radius of a circle.

3. In f(x) = 2x + 3, x is a variable that can take different values to produce different outputs.

2. Types of Mathematical Expressions

An expression is a combination of variables, numbers, and operations. Unlike equations, expressions do not contain equal signs.

2.1 Numerical Expressions

Expressions that contain only numbers and operations.

Examples:

• 5 + 3 × 4

• 7² - 10 ÷ 2

• √16 + 8

2.2 Algebraic Expressions

Expressions that contain at least one variable.

Examples:

• 3x + 5

• y² - 4y + 7

• 2a + 3b - c

Type of Expression	Description	Example
Monomial	Single term with variable(s)	5x, 3y², 7xy
Binomial	Two terms with variables	a + b, 3x - 7
Trinomial	Three terms with variables	x² + 5x + 6
Polynomial	One or more terms with variables	x³ + 4x² - 2x + 5
Rational Expression	Ratio of two polynomials	(x + 3)/(x - 1)

3. Evaluating Expressions

Evaluating an expression means substituting values for variables and calculating the result.

Steps to Evaluate an Expression:

Replace each variable with its given value
Follow the order of operations (PEMDAS/BODMAS)
Perform all calculations

PEMDAS/BODMAS: The order of operations is crucial when evaluating expressions.

PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

BODMAS: Brackets, Orders (powers/roots), Division/Multiplication, Addition/Subtraction

Example 1: Evaluate 3x² - 2x + 5 when x = 4

Step 1: Replace x with 4

3(4)² - 2(4) + 5

Step 2: Calculate according to PEMDAS

3(16) - 2(4) + 5

= 48 - 8 + 5

= 45

Example 2: Evaluate 2a + 3b when a = 5 and b = -2

Step 1: Replace variables with their values

2(5) + 3(-2)

Step 2: Calculate

= 10 + (-6)

= 10 - 6

= 4

4. Simplifying Expressions

Simplifying an expression means rewriting it in an equivalent form with fewer terms or operations.

4.1 Combining Like Terms

Like terms have the same variables raised to the same powers.

Example: Simplify 3x + 5 + 2x - 7

Group like terms: (3x + 2x) + (5 - 7)

= 5x - 2

4.2 Using Properties of Operations

Property	Description	Example
Commutative	Changing order doesn't affect result	a + b = b + a a × b = b × a
Associative	Grouping doesn't affect result	(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)
Distributive	Multiplication distributes over addition	a(b + c) = ab + ac

Example: Simplify 2(3x + 4)

Using the distributive property:

2(3x + 4) = 2(3x) + 2(4)

= 6x + 8

4.3 Simplifying Expressions with Exponents

Rule	Example
x^a × x^b = x^a+b	x² × x³ = x⁵
x^a ÷ x^b = x^a-b	x⁷ ÷ x³ = x⁴
(x^a)^b = x^a×b	(x²)³ = x⁶
(xy)^a = x^ay^a	(2y)³ = 2³y³ = 8y³

Example: Simplify (3x²y)(2xy³)

= 3 × 2 × x² × x × y × y³

= 6 × x³ × y⁴

= 6x³y⁴

5. Translating Words to Expressions

A crucial skill in mathematics is converting word problems into algebraic expressions.

Word/Phrase	Operation	Example
Sum, plus, increased by, more than	Addition (+)	"The sum of x and 5" = x + 5
Difference, minus, decreased by, less than	Subtraction (-)	"x less than 10" = 10 - x
Product, times, multiplied by, of	Multiplication (×)	"The product of 3 and y" = 3y
Quotient, divided by, per, ratio	Division (÷)	"The quotient of z and 4" = z ÷ 4 = z/4