Problem 45 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. (Lesson 1.3) $$6 x-1\quad when\quad x=1$$

Step-by-Step Solution

Answer

The evaluated expression is 5.

Step 1: Identification of the Expression and Substitution Value

The given expression is \(6x - 1\), and the value for \(x\) that we will substitute into this expression is 1.

Step 2: Substitution

Replace the variable \(x\) in the expression with the value 1. After the substitution, the expression becomes \(6(1) - 1\).

Step 3: Evaluation of the Expression

Now, perform the arithmetic operation. The multiplication should be performed first according to the order of operations (BIDMAS/BODMAS), which gives us \(6 - 1\). Then, perform the subtraction, which results in 5.

Key Concepts

Substitution in Algebra

Making sense of algebraic expressions starts with the concept of substitution. This process involves replacing a variable with its corresponding numerical value.

For example, given an expression like \(6x - 1\), substitution comes into play when we are provided with a specific value for \(x\). If \(x\) is given to be 1, we replace every instance of \(x\) in the expression with 1, transforming the expression to \(6(1) - 1\).

Useful steps for substitution include:

• Identify the variable within the expression.
• Determine the value assigned to the variable.
• Replace the variable with the given numerical value in the expression.

It's essential to perform this step correctly to ensure the subsequent arithmetic operations yield the correct result.

Order of Operations

The order in which we perform mathematical operations is crucial to arriving at the right answer. This sequence is often remembered by the acronym PEMDAS/BIDMAS/BODMAS, representing Parentheses/Brackets, Exponents/Indices, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the context of our algebraic expression \(6(1) - 1\), we follow these rules to decide that multiplication must come before subtraction. Hence, we first calculate \(6 \times 1\) to get 6 and then subtract 1, resulting in the final answer of 5.

Incorrectly prioritizing operations can lead to vastly different and incorrect results, emphasizing the importance of this systematic approach.

Arithmetic Operations in Algebra

Arithmetic operations like addition, subtraction, multiplication, and division serve as the foundational building blocks in algebra. When these operations are applied within algebraic expressions, they follow the same principles as they do in basic arithmetic but are combined with the variable components.

In our example, after substituting \(x\) with 1, we are left with arithmetic operations: \(6 \times 1\) and then \(6 - 1\). Because multiplication is done before subtraction according to the order of operations, we find that the given expression simplifies to the numerical answer of 5.

Keeping the hierarchy of operations in mind and executing them correctly within the framework of algebra allows for the accurate simplification and evaluation of such expressions.