Problem 83 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$ \frac{9}{b} \text { when } b=4 $$

Step-by-Step Solution

Answer

The value of the expression \( \frac{9}{b} \) when \( b = 4 \) is 2.25.

Step 1: Understand the problem

The exercise provides an expression \( \frac{9}{b} \). The task is to evaluate or find the value of this expression when \( b = 4 \). This requires substitution.

Step 2: Substitute value

Substitute b = 4 in the given expression. So, replace b in \( \frac{9}{b} \) with 4 to get \( \frac{9}{4} \).

Step 3: Calculate

Perform the division operation. This gives the value \( \frac{9}{4} = 2.25 \).

Key Concepts

Substitution in Algebra

To grasp algebraic expressions, incorporating an understanding of substitution is crucial. Substitution is the process of replacing a variable in an algebraic expression with its numerical value. For instance, in the provided exercise, the expression was \( \frac{9}{b} \) with the given variable being \( b=4 \).

In this case, substitution involves taking the value of \( b \) and inserting it into the expression in the place of the variable. Here's the procedural breakdown:

• Locate the variable within the expression—here, it's the \( b \) in the denominator.
• Replace the variable with its given value—in this context, swap \( b \) with \( 4 \).
• Proceed to evaluate the expression post-substitution, leading to the new expression \( \frac{9}{4} \).

Effectively utilizing substitution helps unravel the complexities of algebra and makes solving problems a systematic process.

Algebraic Expression Evaluation

Algebraic expression evaluation is another fundamental skill in understanding algebra. Essentially, it entails calculating the value of an expression once all the variables are replaced by their numerical equivalents through substitution—as in the previous section.

Once the substitution is made, the next step is to apply the rules of arithmetic operations, such as addition, subtraction, multiplication, and division, to ascertain the numerical value of the expression. In the context of our exercise, after substituting the variable \( b \) with 4, the expression becomes \( \frac{9}{4} \). Evaluating this requires division, the result of which is \( 2.25 \). This step-by-step approach—substituting and then evaluating—is reliable for solving algebraic expressions.

Division in Algebra

Finally, understanding division in algebra is vital—it's one of the essential operations needed to evaluate algebraic expressions. In the context of our exercise example, after substituting \( b \) with 4, we're left with the expression \( \frac{9}{4} \). Division, fundamentally, is the operation of determining how many times one number is contained within another.

Here's how you deal with division in such an algebraic expression:

• First, ensure that the division is the appropriate operation to apply based on the expression's structure (a numerator divided by a denominator).
• Then, carry out the division as you would with regular numbers, \( 9 \) divided by \( 4 \) in our example.
• The result is a decimal \( 2.25 \) if expressed in decimal form or it can stay as a fraction \( \frac{9}{4} \).

Recognizing when and how to divide within an algebraic framework is essential for accurate evaluation of expressions.