Problem 97 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$6 b-39+c \text { when } b=15 \text { and } c=2$$

Step-by-Step Solution

Answer

The value of the expression \(6 b - 39 + c\) when \(b = 15\) and \(c = 2\) is 53.

Step 1: Replace Variables with Given Values

In the given expression \(6 b - 39 + c\), replace the variable 'b' with 15 and 'c' with 2, which results in the expression: \(6(15) - 39 + 2\)

Step 2: Perform Operations in Correct Order

Using the order of operations BIDMAS/BODMAS (Brackets, Indices/Orders, Division/Multiplication, Addition/Subtraction), first perform the multiplication: \(6*15 = 90\). Thus, the expression becomes: \(90 - 39 + 2\)

Step 3: Simplify the Expression

Continue with the addition and subtraction from left to right: \(90 - 39 + 2 = 51 + 2 = 53\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial for evaluating any algebraic expression correctly. By knowing when to perform multiplication, division, addition, and subtraction, we ensure accurate results. This process is guided by the BIDMAS/BODMAS acronym, which stands for:

• Brackets
• Indices (or Orders, which include exponents and roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's break it down with an example from our expression, \(6(15) - 39 + 2\):
First, you start with multiplication, which is higher in the hierarchy than addition and subtraction. So, you multiply \(6\times15\) to get \(90\).
Once multiplication is complete, perform addition and subtraction in order from left to right. Therefore, subtract \(39\) from \(90\), then add \(2\), resulting in the final answer \(53\).
Following BIDMAS/BODMAS accurately prevents mistakes and ensures the expression is evaluated correctly.

Variable Substitution

Variable substitution is the step where variables in an expression are replaced with specific numerical values. This is done to simplify the expression to a form that we can compute numerically. In the given exercise, we are replacing \(b\) with \(15\) and \(c\) with \(2\) in the expression \(6b - 39 + c\).
This substitution transforms the expression to \(6(15) - 39 + 2\).
Here's why variable substitution is important:

• It turns abstract algebraic expressions into concrete numbers, allowing us to perform calculations.
• Makes use of real values in problem-specific situations, providing results that reflect actual scenarios.

Make sure each variable is correctly identified and substituted to avoid errors in further calculations. This step is foundational, as it sets the stage for correctly applying the order of operations.

Simplification in Algebra

Simplification in algebra involves reducing expressions to their simplest form. This is the final step after using the order of operations and substituting variables.
In our example, after substituting and applying the order of operations, the expression becomes \(90 - 39 + 2\).
The goal of simplification is to:

• Make the expression as manageable as possible, both on paper and in the mind.
• Reach a final numerical value which is easy to interpret.

This process involves performing each arithmetic operation in sequence from left to right. Here, subtract \(39\) from \(90\), resulting in \(51\), then add \(2\) to get \(53\).
Effective simplification means checking each step to ensure correctness and that the simplest form is achieved without altering the expression's original values.