Problem 20 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$27-\frac{24}{b} \text { when } b=8$$

Step-by-Step Solution

Answer

The result of the expression when \(b=8\) is \(24\).

Step 1: Substitute the value

Substitute the given value into the formula. For this case, \(b=8\). Substitute \(8\) for \(b\) in the expression \(27-\frac{24}{b}\), to get \(27-\frac{24}{8}\).

Step 2: Perform the Division

Perform the division operation first following the BIDMAS/BODMAS/PEDMAS rules, which prioritizes Division and Multiplication before Addition and Subtraction. This gives \(27-3\).

Step 3: Perform the Subtraction

Now perform the subtraction operation to obtain the value of the expression. This gives \(24\).

Key Concepts

Substitution

Substitution is a step used to simplify algebraic expressions by replacing variables with their given values. This step helps in transforming an expression into a form that can be easily evaluated. For example, in the expression \(27 - \frac{24}{b}\), when you are told \(b = 8\), substitution involves replacing \(b\) with \(8\). Thus, the expression becomes \(27 - \frac{24}{8}\).

• Substitution makes calculations straightforward by turning variables into numbers.
• It helps in understanding how changes in variable values affect the overall expression.

When performing substitution, always ensure to substitute all occurrences of the variable and double-check for any errors. Correct substitution is crucial as it sets the foundation for further operations.

Order of Operations

The order of operations is a vital concept in mathematics, ensuring that expressions are evaluated in a consistent manner. Typically, we use the acronym BIDMAS/BODMAS or PEDMAS standing for:

• Brackets
• Indices (or Orders)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In our example, once \(b\) is substituted by \(8\), the expression \(27 - \frac{24}{8}\) must be evaluated in the correct order. According to BIDMAS/BODMAS/PEDMAS, division is prioritized before subtraction. Thus, you would first calculate \(\frac{24}{8}\), which equals \(3\), transforming the expression to \(27 - 3\).Following these rules helps eliminate mistakes and provides the correct solution to every mathematical expression.

Division

Division is one of the four basic operations in arithmetic, used to determine how many times a number is contained within another number. It is crucial to get it right, as it directly affects the outcome of an expression.In the expression \(27 - \frac{24}{b}\) where \(b = 8\), first substitute the value, giving \(27 - \frac{24}{8}\). Then performing the division \(\frac{24}{8}\) shows \(24\) is split into \(8\) a total of \(3\) times.

• Understand what division represents: distributing or dividing items into equal parts.
• Ensure accuracy by rechecking if the operation results match expectations, especially in complex expressions.

Once the division is calculated, the remaining operations (subtraction, in this case) become simpler and thus ensures you navigate through solving expressions efficiently.