Problem 11 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when \(x=3.\) $$ x^{2}-5 $$

Step-by-Step Solution

Answer

The solution for the expression \(x^{2}-5\) when \(x=3\) is 4.

Step 1: Plugging in Value

The first step is to substitute the value of \(x=3\) into the given expression. Hence, \(x^{2}-5\) becomes \(3^{2}-5\)

Step 2: Carrying out the Evaluations

Now we need to simplify the expression by performing the arithmetic. According to the order of arithmetic operations (BIDMAS/BODMAS), we perform power before subtraction. Hence simplifying \(3^{2}\) we get 9. Now from this result minus 5, to get \(9-5\) which equals 4.

Key Concepts

Substitution in Algebra

Substituting values into algebraic expressions is a fundamental skill in mathematics. It involves replacing variables with specific numbers and then simplifying the expression to obtain a numerical result. For instance, given the expression \(x^2 - 5\), if we are told that \(x=3\), we need to replace every instance of \(x\) in the expression with 3. This process can be visualized as directly swapping the variable for the given value:

Original expression: \(x^2 - 5\)
After substitution: \(3^2 - 5\)

This substitution lays the groundwork for further simplification and solving of the expression, facilitating a clearer understanding of the variable's role within the algebraic expression.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which the operations within an expression should be carried out to ensure accurate results. This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BIDMAS/BODMAS in some regions (Brackets, Indices/Powers, Division and Multiplication, Addition and Subtraction).

In our example with the substituted values, \(3^2 - 5\), we follow these rules by calculating the exponentiation \(3^2\) before carrying out the subtraction. By doing so, we firstly evaluate \(3^2\) to get 9 and only then subtract 5, which orderly gives us the correct result of 4. The order of operations ensures that mathematical expressions are interpreted and simplified consistently without ambiguity.

Simplifying Expressions

Simplifying expressions is the process of breaking down and consolidating an algebraic expression into the simplest form possible. This means performing all the possible calculations, like addition, subtraction, multiplication, division, and exponentiation, while following the order of operations. Our objective is to arrive at a single numerical value or a simpler expression when variables remain.

In the expression \(3^2 - 5\), after substituting \(x\) with 3, through simplification we perform the exponentiation (squared operation) and then the subtraction. Simplification helps in understanding the structure of the expression and may reveal properties that are not initially obvious. Moreover, a simplified expression is easier to work with; it's neater and often less prone to errors in further calculations.