5.1 Algebraic Expressions

Algebraic expressions are the building blocks of math, using letters and symbols to represent numbers and operations. They're like a secret code that lets us describe relationships between quantities and solve complex problems.

Simplifying expressions and performing operations are key skills for working with algebraic expressions. By combining like terms, following the order of operations, and applying basic math rules, we can manipulate these expressions to find solutions and make calculations easier.

Algebraic Expressions

Translation of verbal to algebraic expressions

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• Variables represent unknown or changing quantities in mathematical expressions
  • Common variables include $x$, $y$, $a$, $b$, etc. ($z$, $t$, $n$)
• Mathematical operations express relationships between quantities
  • Addition combines quantities using the $+$ symbol ($5 + 3$)
  • Subtraction finds the difference between quantities using the $-$ symbol ($8 - 2$)
  • Multiplication scales quantities using the $\times$ symbol or parentheses $(2 \times 4$ or $2(4))$
  • Division splits quantities using the $\div$ symbol or a fraction bar ($10 \div 2$ or $\frac{10}{2}$)
• Translating verbal descriptions to algebraic expressions involves identifying the unknown quantity, assigning a variable, and expressing the relationship using appropriate mathematical operations
  • "The difference between a number and 7" translates to $x - 7$, where $x$ represents the unknown number
  • "Three times the sum of a number and 4" translates to $3(x + 4)$, where $x$ represents the unknown number
• Algebraic notation uses symbols and letters to represent numbers and operations in mathematical expressions

Simplification of algebraic expressions

• Like terms have the same variables raised to the same powers and can be combined by adding or subtracting their coefficients
  • $2x^2$ and $-5x^2$ are like terms, while $3y$ and $4y^2$ are not
  • Combining like terms: $4a + 3a = 7a$, $-2b - 5b = -7b$
• The order of operations (PEMDAS) specifies the sequence in which to simplify expressions
  1. Parentheses: Simplify expressions inside parentheses first $(2 + (3 \times 4)) = (2 + 12) = 14$
  2. Exponents: Evaluate exponents, including powers and roots $(2^3 \times 4) = (8 \times 4) = 32$
  3. Multiplication and Division: Perform from left to right $(10 \div 2 \times 3) = (5 \times 3) = 15$
  4. Addition and Subtraction: Perform from left to right $(5 + 3 - 2) = (8 - 2) = 6$
• Simplifying expressions using PEMDAS involves applying each step in the correct order
  • $3 + 2 \times (5 - 2)^2 - 1 = 3 + 2 \times 3^2 - 1 = 3 + 2 \times 9 - 1 = 3 + 18 - 1 = 20$

Operations with algebraic expressions

• Addition and subtraction involve combining like terms and distributing negative signs
  • $(5x - 3) + (2x + 4) = 7x + 1$
  • $(4a + 2b) - (3a - b) = a + 3b$
• Multiplication uses the distributive property to multiply each term of the first expression by each term of the second
  • $(3x - 2)(2x + 1) = 6x^2 + 3x - 4x - 2 = 6x^2 - x - 2$
  • $(a + b)(a - b) = a^2 - b^2$
• Division involves dividing each term of the numerator by the denominator and simplifying the result
  • $\frac{10x^2 - 5x}{5x} = 2x - 1$
  • $\frac{6a^2b + 9ab^2}{3ab} = 2a + 3b$

Mathematical Symbols and Equations

• Mathematical symbols are used to represent operations, relationships, and quantities in algebraic expressions
  • Equality symbol (=) indicates that two expressions have the same value
  • Inequality symbols (<, >, ≤, ≥) show the relationship between two expressions
• An equation is a mathematical statement that asserts the equality of two expressions
  • Equations often involve variables and are used to solve for unknown values