Evaluate an Expression
Pre-AlgebraEvaluating an expression means substituting specific values for variables and calculating the resulting numerical value.
Definition
To evaluate an expression, you replace the variable with a given number and then calculate the answer.
Example
Evaluate $4x + 3$ when $x = 5$. Replace $x$ with $5$: $4(5) + 3 = 20 + 3 = 23$.
Key Insight
Evaluating an expression is like following a recipe. The variable is an ingredient, and the given value tells you how much of that ingredient to use.
Definition
Evaluating an expression means substituting given numerical values for all variables and then simplifying using the order of operations to produce a single number.
Example
Evaluate $3x^2 - 2y + 1$ for $x = 4$ and $y = -3$. Substitute: $3(16) - 2(-3) + 1 = 48 + 6 + 1 = 55$.
Key Insight
Careful substitution into the order of operations (PEMDAS) is essential. A common error is forgetting to apply the exponent before multiplying by the coefficient.
Definition
Evaluation is the map from expressions to values given a variable assignment (a valuation or interpretation). Formally, if $E$ is an expression over variables $x_1, \ldots, x_n$ and $v$ is a valuation mapping each $x_i$ to a value in domain $D$, then $\mathrm{eval}(E, v)$ is the value of $E$ under $v$. In lambda calculus, evaluation is beta reduction of an applied lambda term.
Example
For the polynomial $p(x, y) = x^3 - 2xy + y^2$, the evaluation at the point $(2, 3)$ is $p(2, 3) = 8 - 12 + 9 = 5$. This is the same as computing the dot product of the coefficient vector with the monomial evaluation vector.
Key Insight
Evaluating polynomials efficiently uses Horner's method: $p(x) = a_0 + x(a_1 + x(a_2 + \ldots + xa_n))$ reducing $n^2$ multiplications to $n$, important for numerical computing.