Evaluating The Expression 3(x + 2y) - 2x + 10 When X = 1 And Y = 2

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Evaluating algebraic expressions is a fundamental skill in mathematics. It involves substituting given values for variables and then simplifying the expression using the order of operations. In this article, we will walk through the process of evaluating the expression $3(x + 2y) - 2x + 10$ when $x = 1$ and $y = 2$. This step-by-step guide aims to provide clarity and build a solid understanding of the evaluation process.

Step 1: Substitute the Given Values

The first step in evaluating any algebraic expression is to substitute the given values for the variables. In this case, we are given that $x = 1$ and $y = 2$. We will replace every instance of $x$ in the expression with 1, and every instance of $y$ with 2. This gives us:

3(1+2(2))−2(1)+103(1 + 2(2)) - 2(1) + 10

This substitution sets the stage for simplifying the expression using the order of operations, which ensures we arrive at the correct result. The importance of accurate substitution cannot be overstated, as it forms the foundation for the rest of the evaluation process. Let’s delve deeper into why this initial step is so crucial and how to ensure accuracy.

The Importance of Accurate Substitution

Accurate substitution is the bedrock of evaluating algebraic expressions. Even a minor error in this initial step can cascade through the rest of the calculation, leading to a completely incorrect answer. Think of it as the foundation of a building – if the foundation is flawed, the entire structure is compromised. In our expression, misplacing a 1 or a 2, or accidentally swapping them, will throw off the subsequent steps.

For instance, if we mistakenly substituted $x = 2$ instead of $x = 1$, the expression would become: $3(2 + 2(2)) - 2(2) + 10$, which will lead to a different outcome. Similarly, if we incorrectly substitute $y = 1$ instead of $y = 2$, we would have: $3(1 + 2(1)) - 2(1) + 10$, again altering the final result. These seemingly small errors can have significant consequences.

Best Practices for Substitution

To ensure accuracy in substitution, consider the following best practices:

  1. Double-Check: Always double-check your substitutions immediately after making them. It’s a simple step that can prevent many errors. Compare the original expression and the substituted expression side-by-side to ensure every variable has been correctly replaced with its value.
  2. Use Parentheses: When substituting, especially with negative numbers or more complex expressions, use parentheses. This helps maintain the correct order of operations and avoids sign errors. In our case, using parentheses around the substituted values of $x$ and $y$ helps to visually separate the terms and maintain clarity.
  3. Write Clearly: Ensure your writing is legible. Ambiguous numbers or symbols can lead to misinterpretations. For instance, a poorly written 1 might look like a 7, or a plus sign might resemble a multiplication sign. Clear handwriting minimizes these risks.
  4. Stay Organized: Keep your work organized and methodical. Write each step neatly and in a logical order. This not only helps you track your progress but also makes it easier to spot mistakes when reviewing your work. Use separate lines for each step to enhance clarity.
  5. Focus and Minimize Distractions: When performing substitutions, concentrate on the task at hand. Minimize distractions to reduce the likelihood of making careless errors. A focused approach ensures that you are less likely to overlook critical details.

By following these practices, you can significantly reduce the chances of making errors during substitution. This initial step is the key to unlocking the correct evaluation of the expression, setting the stage for accurate simplification in the subsequent steps.

Step 2: Apply the Order of Operations (PEMDAS/BODMAS)

Once the values have been correctly substituted, the next step is to simplify the expression. The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Adhering to this order is crucial for obtaining the correct result.

Following PEMDAS/BODMAS, we first address the parentheses in the expression: $3(1 + 2(2)) - 2(1) + 10$. Within the parentheses, we have both addition and multiplication. According to the order of operations, multiplication takes precedence over addition. Therefore, we multiply 2 by 2 first:

2(2)=42(2) = 4

Now, the expression within the parentheses becomes:

1+41 + 4

Next, we perform the addition inside the parentheses:

1+4=51 + 4 = 5

Our expression now looks like this:

3(5)−2(1)+103(5) - 2(1) + 10

With the parentheses simplified, we move on to the next operations in the order, which are multiplication and division. In our expression, we have two multiplication operations: $3(5)$ and $2(1)$. We perform these from left to right:

3(5)=153(5) = 15

2(1)=22(1) = 2

Now our expression simplifies to:

15−2+1015 - 2 + 10

Finally, we perform the remaining operations, which are addition and subtraction. These are done from left to right. First, we subtract 2 from 15:

15−2=1315 - 2 = 13

Then, we add 10 to the result:

13+10=2313 + 10 = 23

Thus, by meticulously following the order of operations, we have simplified the expression step-by-step. Each operation was performed in the correct sequence, ensuring that we arrive at the accurate final result. Understanding and applying PEMDAS/BODMAS correctly is vital for simplifying any mathematical expression effectively.

PEMDAS/BODMAS in Detail

To fully grasp the importance of the order of operations, let's break down each component of PEMDAS/BODMAS and discuss why they are sequenced in this particular order:

  1. Parentheses/Brackets: Parentheses (or brackets) are always addressed first. This is because they group terms together, indicating that the operations within them should be treated as a single unit. Simplifying within parentheses ensures that the terms are correctly combined before interacting with terms outside the parentheses. This grouping is essential for maintaining the mathematical integrity of the expression.

  2. Exponents/Orders: Exponents (or orders) represent repeated multiplication and are addressed next. Exponents determine how many times a number is multiplied by itself, so they must be calculated before multiplication or division to maintain the correct magnitude of the terms. Failing to address exponents in the proper order can lead to significant errors in the final result.

  3. Multiplication and Division: Multiplication and division are performed from left to right. These operations have equal precedence, so the order in which they appear in the expression determines the sequence in which they are carried out. This left-to-right rule ensures consistency in calculations and prevents ambiguity.

  4. Addition and Subtraction: Addition and subtraction are the final operations to be performed, also from left to right. Like multiplication and division, they have equal precedence. Performing them in the correct sequence ensures that the terms are combined accurately, leading to the correct final value.

Common Mistakes and How to Avoid Them

Despite the clear rules of PEMDAS/BODMAS, mistakes can still occur. Here are some common errors and strategies to avoid them:

  • Forgetting the Order: One of the most common mistakes is simply forgetting the correct order of operations. To combat this, always write down PEMDAS/BODMAS at the top of your work and check it before each step. This serves as a visual reminder and helps keep you on track.
  • Incorrectly Simplifying Parentheses: Errors within parentheses can easily throw off the entire calculation. Double-check each operation inside the parentheses before moving on. Ensure you've addressed multiplication and division before addition and subtraction within the parentheses.
  • Left-to-Right Confusion: Remember that multiplication and division, as well as addition and subtraction, are performed from left to right. Skipping this rule can lead to incorrect results. If the expression is complex, consider rewriting it after each operation to maintain clarity.
  • Sign Errors: Negative signs can be a source of confusion. Pay close attention to signs when performing operations, especially when dealing with subtraction. Using parentheses to separate terms can help avoid sign errors.
  • Rushing Through Steps: Speed can sometimes lead to errors. Take your time and work through each step carefully. Rushing increases the likelihood of overlooking a critical detail or making a simple mistake.

By understanding the order of operations and being mindful of common mistakes, you can significantly improve your accuracy in simplifying mathematical expressions. PEMDAS/BODMAS is not just a set of rules; it’s a framework for logical problem-solving in mathematics.

Step 3: State the Final Answer

After performing all the necessary operations in the correct order, we arrive at the final result. In our case, the expression $3(1 + 2(2)) - 2(1) + 10$ simplifies to 23. Therefore, the final answer is 23.

This concluding step is essential for clearly communicating the solution. It provides a definitive answer to the problem, leaving no room for ambiguity. Stating the final answer explicitly underscores the entire evaluation process, making it easy for anyone to understand the outcome. Let's delve deeper into why this final step is crucial and how to articulate the result effectively.

The Significance of a Clear Final Answer

A clear final answer is the culmination of all the steps taken to solve a mathematical problem. It serves as the definitive response, encapsulating the result of the entire process. Without a clearly stated final answer, the work done may seem incomplete or inconclusive. Think of it as the conclusion of a well-written essay – it ties everything together, providing closure and clarity.

In our example, after diligently substituting the values, applying the order of operations, and simplifying the expression, stating “The final answer is 23” provides a clear and concise solution. This leaves no doubt about the outcome and ensures that the problem has been fully addressed.

Best Practices for Stating the Final Answer

To ensure that the final answer is communicated effectively, consider the following best practices:

  1. Explicitly State the Answer: Use clear and direct language to state the answer. Phrases like “The final answer is…” or “Therefore, the value of the expression is…” leave no room for misinterpretation. Avoid ambiguous language or incomplete statements.
  2. Highlight the Answer: Consider highlighting the final answer in some way. This can be done by boxing it, underlining it, or using a different color ink. Highlighting makes the answer stand out and easy to identify, especially in a longer solution.
  3. Include Units if Applicable: If the problem involves units (e.g., meters, seconds, dollars), make sure to include the appropriate units in the final answer. This provides context and ensures that the answer is meaningful. For instance, if evaluating an expression that represents a distance, the final answer should include the unit of distance (e.g., 23 meters).
  4. Double-Check the Answer: Before stating the final answer, take a moment to double-check your work. Review each step to ensure that no errors were made. This is a crucial step in preventing mistakes and ensuring the accuracy of the solution.
  5. Relate the Answer to the Problem: If the problem has a real-world context, briefly relate the answer back to the original question. This helps to connect the mathematical solution to the practical situation. For example, if the problem asks for the total cost, the final answer should be stated in terms of cost (e.g., “The total cost is $23”).

Common Pitfalls to Avoid

While stating the final answer may seem straightforward, there are some common pitfalls to avoid:

  • Leaving the Answer Unstated: One of the most common mistakes is not explicitly stating the final answer. This can happen when students assume their work speaks for itself, but it’s essential to provide a clear conclusion.
  • Ambiguous Statements: Avoid vague or ambiguous statements that don’t clearly indicate the final answer. For instance, writing “23” without any context might leave the reader wondering what that number represents.
  • Incorrect Units: Using the wrong units or omitting units entirely can lead to confusion. Always double-check the units and include them in the final answer if applicable.
  • Premature Conclusion: Avoid stating the final answer before completing all the necessary steps. Jumping to a conclusion without proper justification can result in an incorrect solution.
  • Failure to Double-Check: Not double-checking the answer is a significant pitfall. Reviewing your work can catch errors and ensure that the final answer is accurate.

By following these best practices and avoiding common pitfalls, you can ensure that the final answer is clearly and effectively communicated. This final step is the capstone of the problem-solving process, providing a definitive and accurate solution.

Conclusion

In conclusion, evaluating the expression $3(x + 2y) - 2x + 10$ when $x = 1$ and $y = 2$ involves three critical steps: substituting the given values, applying the order of operations, and stating the final answer. By carefully following these steps, we can confidently determine that the value of the expression is 23. This systematic approach not only ensures accuracy but also enhances understanding of algebraic evaluations. Mastering these techniques is essential for success in mathematics and related fields.