Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

Writing about mathematics will help you to learn mathematics. For all writing exercises in this book, use complete sentences to respond to the questions. Some writing exercises can be answered in a sentence; others require a paragraph or two. You can decide how much you need to write as long as your writing clearly and directly answers the question in the exercise. Standard references such as a dictionary and a thesaurus should be helpful. Explain how to evaluate \(2+5 x\) for \(x=3\)

Step-by-Step Solution

Answer

The value of the expression \(2+5x\) for \(x=3\) is \(17\)

Step 1: Understand the Problem

We need to find the value of the algebraic expression \(2 + 5x\), given that the value of \(x\) is \(3\).\nThis involves substituting the given value of the variable \(x\) into the expression.

Step 2: Substitute the value of x

Substitute \(x = 3\) into the expression \(2 + 5x\). This gives us \(2 + 5 * 3\).

Step 3: Evaluate the Expression

Next, we perform the multiplication before the addition due to the BODMAS rule, which gives the sequence of operations as Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right). So \(2 + 5 * 3\), becomes \(2 + 15\).

Step 4: Final Calculation

Finally, we now add \(2 + 15\) to get \(17\), as the evaluation of the expression \(2 + 5x\) when \(x = 3\).

Key Concepts

Algebraic Expression Substitution

When working with algebraic expressions, you'll often encounter variables, which are symbols that represent unknown or changeable numbers. Substituting specific values for these variables is a fundamental skill in algebra. It is exactly like replacing a placeholder in a sentence with the actual word that fits.

For example, in the expression given in our exercise, \(2 + 5x\), the variable \(x\) appears as part of a mathematical phrase. To evaluate the expression for \(x=3\), we literally swap the variable \(x\) with the number \(3\). The expression then transforms from \(2 + 5x\) to \(2 + 5 \times 3\), which is a numerical expression that can be calculated.

Remember, the substitution is like following a recipe - just as you might replace 'sugar' with 'honey', you replace the variable with its value. However, be cautious and substitute numbers carefully to avoid any errors in further steps of calculation. If an expression has more than one instance of the same variable, make sure you substitute each occurrence consistently to preserve the integrity of the expression.

Order of Operations

Understanding the proper sequence of operations in mathematical expressions is crucial and universally recognized by the acronym PEMDAS in the United States. Elsewhere it is known as BODMAS or BIDMAS.

Both acronyms stand for:

• Parentheses/Brackets
• Exponents/Indices
• Multiplication & Division (from left to right)
• Addition & Subtraction (from left to right)

This order is important because performing operations out of sequence can lead to incorrect results. A simple way to remember this is to think of a hierarchy or ranking, just like in a competition: parentheses have first priority, followed by exponents, and so on.

Let’s apply this order to our example: After substituting \(x\) with \(3\), we get \(2 + 5 \times 3\). According to PEMDAS/BODMAS, we first deal with the multiplication \(5 \times 3\), and then we proceed to the addition of \(2\) to the product. This hierarchy helps us maintain consistency in our calculations, ensuring we all arrive at the same correct answer.

Mathematical Problem-Solving Steps

Mastering a systematic approach to solving mathematical problems can turn complex challenges into manageable tasks. The steps, as illustrated by our example of evaluating \(2 + 5x\) when \(x=3\), involve:

• Step 1: Understand the Problem. This means identifying what the problem is asking and what information has been provided. In our case, it's all about finding the value of an algebraic expression.
• Step 2: Substitute the Value. Replace the variable with its given value across the expression. Consistency is key.
• Step 3: Perform Operations in Order. Apply the rules of PEMDAS/BODMAS to ensure the operations are done in the correct sequence to avoid any errors.
• Step 4: Finalize Your Answer. Simplify the expression to get a single numerical value as an answer.

Following these steps can build a strong foundation in any area of mathematics. It's a blueprint for tackling math problems effectively and is an excellent habit for students to develop for their mathematical journey.