Evaluate (x-5xy)(7+yz) For X=1, Y=-3, And Z=4
In this article, we will walk through the process of evaluating the algebraic expression (x-5xy)(7+yz) given specific values for the variables x, y, and z. This exercise is a fundamental concept in algebra, helping to solidify understanding of variable substitution and order of operations. We will substitute x = 1, y = -3, and z = 4 into the expression and then simplify it step by step, ensuring each operation is performed correctly. This type of problem is common in introductory algebra courses and provides a practical application of basic algebraic principles. Mastering these skills is crucial for tackling more complex mathematical problems in the future. By the end of this guide, you will have a clear understanding of how to approach and solve similar algebraic expressions effectively.
To evaluate the expression (x-5xy)(7+yz) when x = 1, y = -3, and z = 4, we will proceed meticulously, ensuring each substitution and operation is performed correctly. This step-by-step approach is crucial for accuracy and for understanding the underlying algebraic principles. First, we'll substitute the given values into the expression, replacing each variable with its corresponding numerical value. This will transform the algebraic expression into a numerical one, which we can then simplify using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Paying close attention to the signs and the order of operations is essential to arrive at the correct answer. Let’s dive into the substitution process and then move towards simplifying the expression systematically. This methodical approach will not only help us solve this particular problem but will also build a solid foundation for tackling more complex algebraic challenges in the future.
1. Substitute the Given Values
The initial step in evaluating the expression (x - 5xy)(7 + yz) is to substitute the given values for the variables x, y, and z. We are given that x = 1, y = -3, and z = 4. Substituting these values directly into the expression, we replace each variable with its corresponding numerical value. This process transforms the algebraic expression into a numerical one, which we can then simplify. Proper substitution is crucial as any error in this step will propagate through the rest of the calculation, leading to an incorrect final answer. We must be particularly careful with negative signs, ensuring they are correctly placed and accounted for. This substitution step sets the stage for the subsequent simplification process, where we will apply the order of operations to arrive at the final evaluated value. Accurate substitution is the cornerstone of solving algebraic problems, making it essential to master this skill. Let’s proceed with the substitution: Replace x with 1, y with -3, and z with 4 in the expression. This will give us the numerical expression that we can then simplify using the order of operations.
Substituting the values, we get:
(1 - 5(1)(-3))(7 + (-3)(4))
2. Simplify within the Parentheses
The next critical step in evaluating the expression (1 - 5(1)(-3))(7 + (-3)(4)) is to simplify the expressions within the parentheses. According to the order of operations (PEMDAS), we must address parentheses first. This involves performing all the operations contained within each set of parentheses before moving on to other parts of the expression. Inside the first set of parentheses, we have 1 - 5(1)(-3), which requires us to perform the multiplication before the subtraction. Similarly, inside the second set of parentheses, we have 7 + (-3)(4), where again, multiplication takes precedence over addition. These steps are crucial because the order in which we perform operations directly affects the outcome. A mistake in this stage can lead to a completely different final answer. Therefore, meticulous attention to the order of operations within the parentheses is essential. Simplifying within parentheses not only reduces the complexity of the expression but also prepares it for the subsequent steps of the evaluation process. Let’s proceed by performing the multiplications within each set of parentheses and then completing the additions and subtractions as required. This methodical approach ensures we adhere to the order of operations and maintain accuracy throughout the simplification process.
First Parenthesis:
1 - 5(1)(-3) = 1 - (-15) = 1 + 15 = 16
Second Parenthesis:
7 + (-3)(4) = 7 + (-12) = 7 - 12 = -5
Now the expression becomes:
(16)(-5)
3. Perform the Final Multiplication
After simplifying the expressions within the parentheses, we arrive at the simplified form of the original expression: (16)(-5). This step involves performing the final multiplication operation to obtain the numerical value of the expression. Multiplication is the final operation in this particular problem, and it’s crucial to perform it accurately to get the correct result. When multiplying two numbers, especially when one or both are negative, it's important to remember the rules of sign multiplication: a positive number multiplied by a negative number results in a negative number. In this case, we are multiplying 16 by -5, so the result will be negative. Performing this multiplication is straightforward but requires careful attention to the sign. The result of this multiplication will be the final answer to the problem, representing the evaluated value of the original algebraic expression for the given values of x, y, and z. This step is the culmination of all the previous steps, and accuracy here is paramount. Let's proceed with the multiplication to find the final answer.
16 * -5 = -80
After meticulously following each step, from substituting the given values into the original expression to simplifying within parentheses and performing the final multiplication, we have arrived at the final answer. The expression (x - 5xy)(7 + yz), when evaluated at x = 1, y = -3, and z = 4, simplifies to -80. This result represents the numerical value of the expression for the specified values of the variables. The process involved careful substitution, adherence to the order of operations (PEMDAS), and accurate arithmetic calculations. This exercise highlights the importance of precision in algebraic manipulations and the step-by-step approach to solving mathematical problems. Understanding and mastering these fundamental concepts is crucial for success in algebra and more advanced mathematical studies. The final answer, -80, is the culmination of our work, demonstrating the power of algebraic principles in transforming symbolic expressions into concrete numerical values.
In conclusion, we have successfully evaluated the algebraic expression (x - 5xy)(7 + yz) for the given values of x = 1, y = -3, and z = 4. The process involved substituting the values into the expression, simplifying within parentheses, and performing the final multiplication. By following the order of operations (PEMDAS) and paying careful attention to signs, we arrived at the final answer of -80. This exercise not only demonstrates the practical application of algebraic principles but also reinforces the importance of accuracy and methodical problem-solving. Such skills are crucial for success in mathematics and related fields. The ability to evaluate expressions like this is a fundamental building block for more advanced topics in algebra and calculus. Mastering these basic concepts allows for a deeper understanding of mathematical relationships and problem-solving strategies. This exercise serves as a reminder of the power and elegance of algebra in simplifying and quantifying complex expressions, providing a solid foundation for further mathematical exploration and learning.
