2.1 Use the Language of Algebra

Algebraic expressions and equations are the building blocks of algebra. They use variables and symbols to represent unknown quantities and relationships, allowing us to model real-world situations and solve complex problems mathematically.

Understanding the components of expressions, like coefficients and constants, is crucial. Knowing how to simplify expressions using exponent rules and the order of operations helps us manipulate these mathematical tools effectively to find solutions.

Algebraic Expressions and Equations

Variables and algebraic symbols

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• Variables represent unknown quantities in mathematical expressions and equations ([object Object],[object Object], [object Object],[object Object], [object Object],[object Object], [object Object],[object Object])
• Algebraic symbols denote mathematical operations or relationships between quantities
  • Addition represented by the plus sign [object Object],[object Object]
  • Subtraction represented by the minus sign [object Object],[object Object]
  • Multiplication represented by the times sign $\times$, a dot $\cdot$, or parentheses $(ab)$
  • Division represented by the division sign [object Object],[object Object] or fraction notation $\frac{a}{b}$
  • Equality represented by the equal sign $=$
• Combine variables and symbols to create algebraic expressions ($3x + 2$) or equations ($2x - 5 = 7$) that model real-world situations and mathematical problems

Expressions vs equations

• Algebraic expressions combine variables, numbers, and operations without an equal sign ($4x$, $2y - 3$, $\frac{x}{2} + 5$)
  • Expressions can be simplified by combining like terms or applying properties of operations
• Algebraic equations state the equality between two expressions using an equal sign ($3x = 12$, $y - 2 = 5$, $\frac{x}{4} + 3 = 7$)
  • Equations can be solved to find the value of the variable that makes the equation true
• Expressions and equations are fundamental tools in algebra for modeling and solving mathematical problems

Components of Algebraic Expressions

• Coefficient: The numerical factor of a term containing a variable (e.g., in $3x^2$, 3 is the coefficient)
• Constant: A term in an expression that has a fixed value and does not contain a variable (e.g., in $2x + 5$, 5 is the constant)
• Like terms: Terms in an expression that have the same variables raised to the same powers (e.g., $3x$ and $5x$ are like terms)
• Distributive property: A property that allows multiplication to be distributed over addition or subtraction (e.g., $a(b + c) = ab + ac$)
• Algebraic fraction: A fraction where the numerator, denominator, or both contain algebraic expressions (e.g., $\frac{x+1}{x-2}$)

Exponent rules for simplification

• Exponents indicate repeated multiplication of a base number ($3^2 = 3 \times 3 = 9$)
• Multiply powers with the same base by adding exponents ($2^3 \times 2^4 = 2^{3+4} = 2^7$)
• Divide powers with the same base by subtracting exponents ($\frac{x^5}{x^2} = x^{5-2} = x^3$)
• Raise a power to another power by multiplying exponents ($(3^2)^4 = 3^{2 \times 4} = 3^8$)
• Negative exponents represent the reciprocal of the base with a positive exponent ($2^{-3} = \frac{1}{2^3} = \frac{1}{8}$)
• Apply these rules to simplify complex algebraic expressions involving exponents

Order of operations in expressions

1. Parentheses: Simplify expressions inside parentheses first
2. Exponents: Evaluate exponents second
3. Multiplication and Division: Perform from left to right
4. Addition and Subtraction: Perform from left to right

• PEMDAS mnemonic helps remember the order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Example: $2 + 3 \times (4^2 - 1) \div 5$
  1. Parentheses: $4^2 - 1 = 16 - 1 = 15$
  2. Exponents: already simplified
  3. Multiplication and Division from left to right: $3 \times 15 \div 5 = 45 \div 5 = 9$
  4. Addition: $2 + 9 = 11$
• Following the correct order of operations ensures accurate evaluation of multi-step algebraic expressions