Evaluate 7(x+1)-5 At X=3 A Step-by-Step Guide
In this article, we will delve into the process of evaluating the algebraic expression 7(x+1)-5 when the variable x is assigned the value of 3. This type of problem is a fundamental concept in algebra and is essential for understanding more complex mathematical operations. We will break down the steps involved in solving this problem, ensuring a clear and comprehensive understanding for anyone, regardless of their mathematical background. Our primary focus will be on substituting the given value of x into the expression and then simplifying the expression using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding how to evaluate algebraic expressions is crucial not only for success in mathematics courses but also for various real-world applications where mathematical models are used to represent and solve problems.
Evaluating algebraic expressions involves replacing variables with specific numerical values and then performing the indicated operations. This process is a cornerstone of algebra and is used extensively in various fields, including science, engineering, and economics. For instance, in physics, you might use an algebraic expression to calculate the distance an object travels given its speed and time. In finance, you might use an expression to calculate the future value of an investment. Therefore, mastering the skill of evaluating expressions is not just an academic exercise but a practical skill with wide-ranging applications. In the following sections, we will go step-by-step through the evaluation of the given expression, highlighting the importance of each step and providing explanations to enhance understanding. We will also discuss common mistakes to avoid and offer tips for solving similar problems more efficiently.
To effectively evaluate the expression 7(x+1)-5 at x=3, we will follow a step-by-step approach that ensures accuracy and clarity. This approach aligns with the standard order of operations, which dictates the sequence in which mathematical operations should be performed. Ignoring this order can lead to incorrect results, so it's crucial to adhere to it strictly. The order of operations is typically remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This mnemonic serves as a roadmap for simplifying mathematical expressions, ensuring that everyone arrives at the same correct answer.
Our first step will be to substitute the value of x, which is 3, into the expression. This means replacing every instance of the variable x in the expression with the number 3. After the substitution, we will have a numerical expression that we can simplify. The next step involves simplifying the expression inside the parentheses. According to PEMDAS, operations within parentheses are always performed first. Once we've simplified the parentheses, we'll move on to the multiplication operation. The number 7 is multiplied by the result of the parentheses simplification. Finally, we'll perform the subtraction operation, subtracting 5 from the result of the multiplication. Each of these steps is crucial, and we will elaborate on them in detail in the subsequent paragraphs, providing clear explanations and justifications for each action. This structured approach will not only help in solving this particular problem but will also serve as a template for tackling similar algebraic evaluations in the future.
1. Substitute the Value of x
The initial step in evaluating 7(x+1)-5 at x=3 is to substitute the given value of x into the expression. This involves replacing every instance of x with the number 3. This substitution transforms the algebraic expression into a numerical expression, which we can then simplify using arithmetic operations. The accuracy of this substitution is paramount, as any error in this step will propagate through the rest of the solution, leading to an incorrect final answer. Careful attention to detail during substitution is therefore essential. After the substitution, the expression will no longer contain the variable x; instead, it will consist only of numbers and arithmetic operations. This sets the stage for the next steps, which involve simplifying the numerical expression according to the order of operations.
Substituting x = 3 into the expression 7(x+1)-5, we get 7(3+1)-5. Notice that we have replaced the variable x with the number 3, maintaining the rest of the expression intact. The parentheses around (3+1) are crucial, as they indicate that this operation should be performed before any others, according to the order of operations. The number 7 is still multiplied by the result of the expression inside the parentheses, and the subtraction of 5 is still pending. This substitution step is a straightforward mechanical process, but it is the foundation upon which the rest of the solution is built. A mistake here can derail the entire process, so it's worth double-checking the substitution to ensure accuracy. With the value of x successfully substituted, we can now move on to the next step, which involves simplifying the expression inside the parentheses.
2. Simplify Inside the Parentheses
Following the order of operations (PEMDAS), after substituting the value of x, the next crucial step in evaluating 7(x+1)-5 at x=3 is to simplify the expression within the parentheses. Parentheses take precedence over other operations, so we must address them before moving on to multiplication, division, or subtraction. In our case, the expression inside the parentheses is (3+1). This is a simple addition operation, but it's essential to perform it correctly to ensure the accuracy of the subsequent steps. The result of this addition will then be used in the next operation, which is multiplication. Simplifying the expression within the parentheses not only follows the rules of PEMDAS but also makes the overall expression easier to manage and less prone to errors. By breaking down the problem into smaller, more manageable parts, we can reduce the chances of making mistakes and increase our confidence in the solution.
To simplify (3+1), we perform the addition operation, which results in 4. So, the expression 7(3+1)-5 becomes 7(4)-5. We have now successfully simplified the expression inside the parentheses, reducing it to a single numerical value. The parentheses themselves are no longer necessary, as the operation within them has been completed. The next operation we need to address, according to PEMDAS, is multiplication. We have the number 7 multiplied by the result of the parentheses, which is 4. This multiplication will be the next step in our evaluation process, bringing us closer to the final answer. By carefully following the order of operations and simplifying each part of the expression systematically, we ensure that our solution is accurate and reliable. This methodical approach is a key skill in algebra and will be valuable in solving more complex problems in the future.
3. Perform Multiplication
With the parentheses simplified, the next step in evaluating 7(x+1)-5 at x=3 is to perform the multiplication. According to the order of operations (PEMDAS), multiplication and division take precedence over addition and subtraction. In our simplified expression, 7(4)-5, the multiplication operation is 7 multiplied by 4. This is a straightforward multiplication, but it's a critical step in arriving at the correct final answer. The result of this multiplication will then be used in the final operation, which is subtraction. Accurate multiplication is essential, as any error here will affect the final result. By performing the multiplication before the subtraction, we adhere to the rules of PEMDAS and maintain the integrity of the mathematical expression.
Performing the multiplication, 7 multiplied by 4 equals 28. So, the expression 7(4)-5 becomes 28-5. We have now completed the multiplication step, reducing the expression to a simple subtraction problem. The number 28 represents the product of 7 and 4, and it will be the starting point for the final operation. The only remaining operation is subtraction, which involves subtracting 5 from 28. This final step will give us the evaluated value of the original expression when x is equal to 3. By systematically working through each operation in the correct order, we have simplified the expression step-by-step, bringing us closer to the solution. The multiplication step is a crucial part of this process, and its accurate execution is vital for obtaining the correct final answer.
4. Perform Subtraction
After completing the multiplication, the final step in evaluating 7(x+1)-5 at x=3 is to perform the subtraction. According to the order of operations (PEMDAS), subtraction is the last operation to be carried out in this expression. We have simplified the expression to 28-5, which is a straightforward subtraction problem. This step will give us the final numerical value of the expression when x is equal to 3. Accurate subtraction is crucial, as it is the last operation and determines the ultimate result. By performing this subtraction, we will have successfully evaluated the entire expression, completing the problem.
Subtracting 5 from 28, we get 23. Therefore, the expression 7(x+1)-5 evaluated at x=3 is equal to 23. This is the final answer to our problem. We have systematically worked through each step, from substitution to simplification, following the order of operations to arrive at this result. The subtraction step was the culmination of all the previous steps, and its accurate execution has allowed us to determine the value of the expression. This process demonstrates the importance of following the correct order of operations and performing each step carefully to ensure an accurate final answer. With this result, we have successfully evaluated the expression and can confidently move on to similar problems in the future.
In conclusion, after systematically evaluating the expression 7(x+1)-5 at x=3, we have arrived at the final answer. By following the order of operations (PEMDAS) and carefully executing each step, we have successfully simplified the expression and determined its value. The process involved substituting the value of x, simplifying the expression within parentheses, performing multiplication, and finally, carrying out the subtraction. Each of these steps was crucial in arriving at the correct result, and any error in any step would have affected the final answer. The final answer represents the numerical value of the expression when x is equal to 3, providing a clear and concise solution to the problem.
The final answer to the evaluation of 7(x+1)-5 at x=3 is 23. This result demonstrates the power of algebraic evaluation and the importance of following the correct order of operations. By understanding and applying these principles, we can confidently solve similar problems and tackle more complex mathematical challenges. The step-by-step approach we used in this article provides a clear and structured method for evaluating algebraic expressions, ensuring accuracy and minimizing the risk of errors. This skill is not only essential in mathematics but also in various other fields where mathematical models are used to solve real-world problems. Therefore, mastering this skill is a valuable asset in both academic and professional pursuits.
