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 |
|---|---|
| xa × xb = xa+b | x2 × x3 = x5 |
| xa ÷ xb = xa-b | x7 ÷ x3 = x4 |
| (xa)b = xa×b | (x2)3 = x6 |
| (xy)a = xaya | (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 |
Example 1: "Seven more than twice a number"
Let n = the number
Twice a number = 2n
Seven more than twice a number = 2n + 7
Example 2: "The quotient of a number and 5, decreased by 3"
Let n = the number
The quotient of a number and 5 = n/5
Decreased by 3 = n/5 - 3
6. Real-world Applications
6.1 Area and Perimeter Formulas
Rectangle: Area = length × width = l × w
Rectangle: Perimeter = 2 × length + 2 × width = 2l + 2w
Circle: Area = π × radius² = πr²
Circle: Circumference = 2 × π × radius = 2πr
6.2 Financial Formulas
Simple Interest: I = P × r × t
Where I = interest, P = principal, r = rate, t = time
Compound Interest: A = P(1 + r)ᵗ
Where A = final amount, P = principal, r = rate, t = time
6.3 Distance, Rate, and Time
Distance = Rate × Time: d = rt
Rate = Distance ÷ Time: r = d/t
Time = Distance ÷ Rate: t = d/r
7. Common Strategies for Solving Expression Problems
7.1 Substitution Method
Replace variables with given values and simplify.
Example: If f(x) = x² - 3x + 2, find f(4)
f(4) = (4)² - 3(4) + 2
= 16 - 12 + 2
= 6
7.2 Grouping and Factoring
Rearrange expressions to identify common factors.
Example: Simplify 3x + 6 + 2x - 4
Group like terms: (3x + 2x) + (6 - 4)
= 5x + 2
7.3 Working Backwards
Start with the answer and work backwards to understand the expression.
Example: If 2(x + 3) = 16, what is x?
2(x + 3) = 16
x + 3 = 16 ÷ 2
x + 3 = 8
x = 5
8. Practice Problems
Problem 1: Evaluate 2x² - 5x + 3 when x = -2
Solution:
2(-2)² - 5(-2) + 3
= 2(4) - 5(-2) + 3
= 8 + 10 + 3
= 21
Problem 2: Simplify 3(2x - 4) + 2(x + 5)
Solution:
3(2x - 4) + 2(x + 5)
= 6x - 12 + 2x + 10
= 8x - 2
Problem 3: Write an expression for "The product of a number and 6, increased by twice the number"
Solution:
Let n = the number
The product of a number and 6 = 6n
Twice the number = 2n
The product of a number and 6, increased by twice the number = 6n + 2n = 8n
Variables and Expressions Quiz
Question 1: Evaluate the expression 3x² - 4x + 2 when x = 3.
3(3)² - 4(3) + 2 = 3(9) - 12 + 2 = 27 - 12 + 2 = 17
Question 2: Simplify the expression: 2(3x + 4) - 5(x - 1)
2(3x + 4) - 5(x - 1) = 6x + 8 - 5x + 5 = x + 13
Question 3: Which expression represents "5 less than twice a number"?
"5 less than twice a number" means we start with twice the number (2n) and then subtract 5, giving us 2n - 5.
Question 4: If f(x) = x² + 3x - 4, what is f(-2)?
f(-2) = (-2)² + 3(-2) - 4 = 4 - 6 - 4 = -6
Question 5: Simplify the expression: (3x²y)(2xy²)
(3x²y)(2xy²) = 3 × 2 × x² × x × y × y² = 6x³y³