Problem 93 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ \frac{n}{2}[2 a+(n-1) d] \text { for } n=10, a=-4.2, \text { and } d=6.6 $$

Step-by-Step Solution

Answer

The expression evaluates to 255.

Step 1: Identify Parameters

First, identify the given values for the variables in the expression. We have \( n = 10 \), \( a = -4.2 \), and \( d = 6.6 \). These values will replace the variables in the given formula.

Step 2: Substitute the Parameters into the Formula

Replace the variables in the formula \( \frac{n}{2}[2 a+(n-1) d] \) with the given values. This results in:\[ \frac{10}{2}[2(-4.2) + (10-1)6.6] \]

Step 3: Simplify Inside the Brackets

Calculate the expressions inside the brackets. Start with:- \( 2 \times (-4.2) = -8.4 \)- \( 10 - 1 = 9 \)- \( 9 \times 6.6 = 59.4 \)Now substitute these back into the bracket:\[ [-8.4 + 59.4] \]

Step 4: Perform Bracket Addition

Now add the results from the previous step:\[ -8.4 + 59.4 = 51 \]

Step 5: Calculate the Final Expression

Calculate the expression outside the brackets:\[ \frac{10}{2} = 5 \]Now multiply this by the result from inside the brackets:\[ 5 \times 51 = 255 \]

Step 6: Conclusion: Write the Final Answer

Upon evaluating the expression with the given values, the result is 255.

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental technique used to simplify expressions or solve equations by replacing variables with given values. It essentially involves taking the provided values for variables and inserting them into an expression or equation to evaluate it. This method helps in transforming abstract mathematical problems into more tangible ones.

Here’s a simple way to think about substitution:

• Identify the variables in the equation or expression.
• Take note of the values given for each variable.
• Replace the variables with these specific values to form a new expression.

In our example, the expression \ \( \frac{n}{2}[2a+(n-1)d] \ \) includes the variables \( n, a, \) and \( d \). The provided values for these variables — \( n = 10 \, a = -4.2, \) and \( d = 6.6 \) — are substituted into the expression. This replaces the theoretical component with numerical values, making it easier to solve. By doing so, the initially complex expression turns into something you can easily calculate step by step.

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical equations into simpler forms, where calculations become more manageable. It’s a necessary step while solving algebraic expressions, ensuring each part of the equation is dealt with correctly.

In the example given, simplification occurs in several steps:

• First, tackle the multiplication within the inner brackets: \( 2 \times (-4.2) = -8.4 \).
• Then, resolve any other necessary operations inside the brackets, such as \((10 - 1) \times 6.6 = 59.4 \).
• After evaluating each component inside the bracket, sum the results to have a single number: \(-8.4 + 59.4 = 51 \).
• Finally, simplify the expression outside the brackets: \( \frac{10}{2} = 5 \).

Ultimately, you piece together these simplified components, making the calculation straightforward. The process removes clutter from the equation and focuses on core calculations, ensuring nothing gets lost in the complexity of multiple terms.

Step-by-Step Problem Solving

Step-by-step problem solving in mathematics involves breaking down a problem into smaller, manageable steps that are easier to understand and solve sequentially. This approach is crucial for handling arithmetic series and complex algebraic expressions. In the context of evaluating expressions like the given one, this structured method helps to avoid errors and ensures clarity.

Here’s how it plays out for our problem:

• Step 1: Identify and note down all given parameters. This sets the groundwork for substitution.
• Step 2: Substitute the known values into the expression. Make sure each variable is replaced accurately.
• Step 3: Begin simplifying by resolving operations inside brackets first, following the order of operations.
• Step 4: Once the brackets are simplified to a single value, manage any remaining operations, such as multiplication outside the brackets.
• Step 5: Finalize your solution by calculating the results displayed after simplification.

By tackling problems one step at a time, students can develop a clear path from the initial complex problem to the simple solution. This step-by-step strategy is not just useful for this exercise but is a transferable skill applicable to all areas of arithmetic and algebra.