Evaluate 2y(3z-x) For X=2 Y=5 And Z=4
In this article, we will delve into the process of evaluating algebraic expressions, specifically focusing on the expression 2y(3z - x). This is a fundamental concept in algebra, essential for solving various mathematical problems and understanding more complex concepts. We will walk through the steps involved, substituting given values for variables and simplifying the expression to arrive at the final answer. Understanding how to evaluate expressions is crucial for anyone studying mathematics, science, or engineering, as it forms the basis for many calculations and problem-solving techniques.
Understanding Algebraic Expressions
Before we dive into evaluating the specific expression, let's first understand what an algebraic expression is. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters like x, y, and z) that represent unknown values. Constants are fixed numerical values.
For instance, in the expression 2y(3z - x), x, y, and z are the variables, and the numbers 2 and 3 are constants. The operations involved are multiplication and subtraction. Evaluating an expression means finding its numerical value by substituting the variables with given values and performing the indicated operations. This process allows us to determine the expression's worth under specific conditions. Mastering the evaluation of algebraic expressions is a key step in progressing to more advanced algebraic topics, such as solving equations and inequalities.
The Given Expression: 2y(3z - x)
Our primary focus is the expression 2y(3z - x). This expression involves three variables (x, y, and z) and several operations. To evaluate this expression, we need to substitute the given values for the variables and then simplify the expression using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the operations in the correct sequence to obtain the accurate result. Ignoring the order of operations can lead to incorrect answers, highlighting the importance of following this rule. The expression showcases how different operations and variables can be combined to form a mathematical statement, and its evaluation demonstrates the practical application of algebraic principles.
Step-by-Step Evaluation
Now, let's proceed with the step-by-step evaluation of the expression 2y(3z - x) for the given values x = 2, y = 5, and z = 4. This process involves careful substitution and simplification to arrive at the final numerical value. Each step is crucial to ensure accuracy, and understanding the logic behind each step is essential for mastering algebraic evaluations. By breaking down the process into smaller, manageable steps, we can minimize errors and gain a deeper understanding of how the expression behaves under different conditions. This detailed approach also helps in developing problem-solving skills that are applicable to a wide range of mathematical problems.
Step 1: Substitute the Values
The first step is to substitute the given values for the variables in the expression. We have x = 2, y = 5, and z = 4. Replacing the variables with their corresponding values, we get:
2(5)(3(4) - 2)
This substitution transforms the algebraic expression into a numerical expression, which we can then simplify using arithmetic operations. This step is the foundation of the evaluation process, as it sets the stage for the subsequent calculations. Accuracy in substitution is paramount, as any errors at this stage will propagate through the rest of the evaluation. The ability to correctly substitute values is a fundamental skill in algebra and is used extensively in various mathematical and scientific applications.
Step 2: Simplify Inside the Parentheses
Following the order of operations (PEMDAS), we need to simplify the expression inside the parentheses first. In our case, the expression inside the parentheses is (3(4) - 2). We start by performing the multiplication:
3(4) = 12
Now, we substitute this result back into the parentheses:
(12 - 2)
Next, we perform the subtraction:
12 - 2 = 10
So, the simplified expression inside the parentheses is 10. This step demonstrates the importance of adhering to the order of operations to ensure accurate simplification. Parentheses indicate a priority in calculations, and simplifying within them first is crucial for obtaining the correct result. The process of simplifying expressions within parentheses is a common task in algebra and is a building block for more complex algebraic manipulations.
Step 3: Perform the Multiplication
Now that we have simplified the expression inside the parentheses, we can proceed with the multiplication. Our expression now looks like this:
2(5)(10)
We perform the multiplication from left to right:
2(5) = 10
Then, we multiply the result by the remaining factor:
10(10) = 100
Therefore, the final result of the evaluation is 100. This step highlights the final stage of simplifying the expression, where the remaining multiplications are carried out to arrive at a single numerical value. The systematic approach to multiplication, following the order of operations, ensures accuracy in the final calculation. The ability to perform multiplication efficiently and accurately is a fundamental skill in mathematics and is essential for various applications.
Final Answer: 100
After performing all the necessary steps, we have successfully evaluated the expression 2y(3z - x) for x = 2, y = 5, and z = 4. The final answer is 100. This result represents the numerical value of the expression under the given conditions. The process of evaluation involves careful substitution and simplification, adhering to the order of operations to ensure accuracy. The final answer provides a concrete value for the expression, which can be used in further calculations or analysis. Understanding how to evaluate expressions is a fundamental skill in algebra and is crucial for solving various mathematical problems.
Importance of Order of Operations
The order of operations (PEMDAS/BODMAS) is crucial in evaluating mathematical expressions correctly. It dictates the sequence in which operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to this order ensures that we arrive at the correct answer.
For example, in the expression 2y(3z - x), we first simplified the expression within the parentheses (3z - x) before performing any multiplication. Ignoring the order of operations can lead to incorrect results. Understanding and applying the order of operations is a fundamental skill in mathematics and is essential for solving equations, simplifying expressions, and performing various calculations accurately. The order of operations provides a standardized approach to mathematical calculations, ensuring consistency and avoiding ambiguity.
Practice and Mastery
Practice is key to mastering the evaluation of algebraic expressions. By working through various examples and problems, you can reinforce your understanding of the concepts and improve your problem-solving skills. Start with simple expressions and gradually move on to more complex ones. Pay attention to the order of operations and be mindful of the signs and symbols.
Regular practice will help you become more confident and efficient in evaluating expressions. It will also enhance your overall mathematical abilities and prepare you for more advanced topics. There are numerous resources available for practice, including textbooks, online tutorials, and practice worksheets. Consistent effort and dedication to practice will lead to mastery and a deeper understanding of algebraic concepts. The ability to confidently evaluate expressions is a valuable skill that will benefit you in various academic and professional pursuits.
Conclusion
In conclusion, evaluating algebraic expressions is a fundamental skill in mathematics. We have successfully evaluated the expression 2y(3z - x) for x = 2, y = 5, and z = 4, arriving at the final answer of 100. This process involved substituting the given values for the variables, simplifying the expression inside the parentheses, and performing the multiplication. We also emphasized the importance of adhering to the order of operations to ensure accuracy.
Mastering the evaluation of algebraic expressions is crucial for various mathematical and scientific applications. Consistent practice and a thorough understanding of the underlying concepts will enable you to confidently solve a wide range of problems. This skill serves as a building block for more advanced algebraic topics and is essential for success in mathematics and related fields.
