Algebraic expressions are mathematical phrases that combine numbers and variables using operations such as addition, subtraction, multiplication, and division. A variable is a letter that represents an unknown or changeable value, commonly denoted by letters like x, y, or z. Unlike variables, a coefficient is a fixed number multiplying the variable, and a constant is a standalone number without any variable attached. For example, in the expression \$2x + 5\(, the number 2 is the coefficient of the variable \)x\(, and 5 is the constant.
Understanding the difference between variables, coefficients, and constants is crucial because while variables can change values, coefficients and constants remain fixed. This distinction helps when working with algebraic expressions, especially when evaluating them for specific values of the variables.
To evaluate an algebraic expression, substitute the given values for the variables and then perform the arithmetic operations following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For instance, evaluating the expression \)2x + 5\( for \)x = 4\( involves replacing \)x\( with 4, resulting in \)2 \times 4 + 5\(. Calculating this gives \)8 + 5 = 13\(.
When an expression contains multiple variables, such as \)\frac{1}{2}a + 4b\(, you substitute each variable with its given value. For example, if \)a = 10\( and \)b = -6\(, the expression becomes \)\frac{1}{2} \times 10 + 4 \times (-6)\(. Simplifying, \)\frac{1}{2} \times 10 = 5\( and \)4 \times (-6) = -24\(, so the expression evaluates to \)5 - 24 = -19\(.
Exponents are also important in algebraic expressions. For example, to evaluate \)-8y^3\( when \)y = 2\(, first calculate the exponent: \)2^3 = 2 \times 2 \times 2 = 8\(. Then multiply by the coefficient: \)-8 \times 8 = -64\(. Thus, the expression evaluates to \)-64$.
Mastering algebraic expressions involves recognizing variables, coefficients, and constants, and confidently substituting values to evaluate expressions. This foundational skill is essential for solving equations and understanding more complex algebraic concepts.