Evaluating Algebraic Expressions A Step By Step Guide

In this comprehensive guide, we will delve into the process of evaluating algebraic expressions. Specifically, we will tackle the expression $\frac{3}{2}y - 3 + \frac{5}{3}z$ given the values $y = 6$ and $z = 3$. This seemingly simple task forms the bedrock of more complex algebraic manipulations and is a crucial skill for anyone venturing into the realm of mathematics, physics, engineering, or any field that relies on quantitative analysis. Understanding how to correctly substitute values and simplify expressions is not just about getting the right answer; it's about developing a logical and methodical approach to problem-solving. We'll break down each step, providing clear explanations and justifications, ensuring that you not only grasp the mechanics but also the underlying principles. Whether you're a student grappling with algebra for the first time or a seasoned professional brushing up on your skills, this guide will provide a solid foundation for your algebraic endeavors. By the end of this exploration, you'll be equipped with the knowledge and confidence to tackle similar problems with ease and precision. So, let's embark on this mathematical journey together, unlocking the secrets of expression evaluation and empowering you to conquer algebraic challenges.

Before we jump into the substitution, let's take a closer look at the expression $\frac{3}{2}y - 3 + \frac{5}{3}z$. This expression is a combination of terms, each contributing to the overall value. The first term, $\frac{3}{2}y$, represents a fraction multiplied by a variable, highlighting the importance of understanding fractional coefficients in algebraic expressions. The coefficient $\frac{3}{2}$ scales the value of $y$, demonstrating the concept of proportionality in mathematical relationships. The second term, $-3$, is a constant, a fixed numerical value that doesn't change regardless of the values of $y$ and $z$. Constants are the anchors of expressions, providing a baseline around which the other terms fluctuate. The third term, $\frac{5}{3}z$, mirrors the structure of the first term, showcasing another fractional coefficient scaling a variable. The coefficient $\frac{5}{3}$ acts similarly to $\frac{3}{2}$, emphasizing the role of fractions in determining the magnitude of the term. Understanding the individual components of an expression is crucial for effective evaluation. By dissecting the expression into its constituent parts, we can approach the substitution and simplification process with clarity and precision. This meticulous approach not only helps in solving the immediate problem but also fosters a deeper understanding of algebraic structures, paving the way for tackling more complex equations and formulas. So, with a clear understanding of each term's role, we are now well-prepared to embark on the next step: substituting the given values.

Now comes the crucial step: substitution. We are given that $y = 6$ and $z = 3$. Substitution, in its essence, is the process of replacing variables with their corresponding numerical values. It's like plugging in the missing pieces of a puzzle to reveal the complete picture. In our expression, $\frac{3}{2}y - 3 + \frac{5}{3}z$, the variables $y$ and $z$ act as placeholders, waiting to be filled with the specified values. When we substitute $y = 6$, we replace the symbol $y$ with the number $6$, effectively transforming the term $\frac{3}{2}y$ into $\frac{3}{2} \times 6$. Similarly, when we substitute $z = 3$, we replace the symbol $z$ with the number $3$, transforming the term $\frac{5}{3}z$ into $\frac{5}{3} \times 3$. It's crucial to perform the substitution accurately, ensuring that each variable is replaced with its correct value. A single mistake in substitution can lead to an incorrect final answer. Think of it as a delicate operation where precision is paramount. Once the substitution is complete, the expression transforms from an abstract symbolic representation into a concrete numerical calculation. This is a pivotal moment in the evaluation process, as we transition from algebra to arithmetic. With the variables now replaced by numbers, we are ready to embark on the simplification phase, where we will use the rules of arithmetic to arrive at the final solution. So, with the substitution successfully executed, we stand on the threshold of simplification, eager to unravel the numerical value hidden within the expression.

With the values substituted, our expression now reads $\frac{3}{2} \times 6 - 3 + \frac{5}{3} \times 3$. This is where the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), comes into play. It dictates the sequence in which we perform the arithmetic operations to ensure a consistent and accurate result. According to PEMDAS, we prioritize multiplication before addition and subtraction. So, we start by evaluating the terms $\frac{3}{2} \times 6$ and $\frac{5}{3} \times 3$. When multiplying a fraction by a whole number, it's often helpful to visualize the whole number as a fraction with a denominator of 1. Thus, $\frac{3}{2} \times 6$ becomes $\frac{3}{2} \times \frac{6}{1}$. To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). This gives us $\frac{3 \times 6}{2 \times 1} = \frac{18}{2}$. Similarly, $\frac{5}{3} \times 3$ becomes $\frac{5}{3} \times \frac{3}{1} = \frac{5 \times 3}{3 \times 1} = \frac{15}{3}$. Now, we simplify these fractions. $\frac{18}{2}$ simplifies to $9$, and $\frac{15}{3}$ simplifies to $5$. Our expression now looks like $9 - 3 + 5$. Next, we perform the addition and subtraction from left to right. $9 - 3$ equals $6$, and then $6 + 5$ equals $11$. Therefore, the final value of the expression is $11$. This step-by-step calculation showcases the importance of following the order of operations and performing each arithmetic operation with precision. By adhering to these principles, we can confidently arrive at the correct answer and ensure the accuracy of our mathematical manipulations.

To solidify our understanding, let's present the solution in a clear, step-by-step format:

1. Original Expression: $\frac{3}{2}y - 3 + \frac{5}{3}z$
2. Substitute Values: $\frac{3}{2}(6) - 3 + \frac{5}{3}(3)$
3. Multiply Fractions: $\frac{18}{2} - 3 + \frac{15}{3}$
4. Simplify Fractions: $9 - 3 + 5$
5. Subtract: $6 + 5$
6. Add: $11$

This step-by-step breakdown provides a clear roadmap of the entire evaluation process. Each step builds upon the previous one, leading us systematically to the final solution. This structured approach not only helps in understanding the mechanics of the problem but also promotes a logical and methodical way of thinking, which is invaluable in mathematics and beyond. By carefully tracing each step, we can identify the key operations and their sequence, reinforcing our grasp of algebraic evaluation. This detailed solution serves as a valuable reference point, allowing us to revisit the process and solidify our understanding. So, with this step-by-step guide, we have a powerful tool for tackling similar problems and confidently navigating the world of algebraic expressions.

In conclusion, we have successfully evaluated the expression $\frac{3}{2}y - 3 + \frac{5}{3}z$ when $y = 6$ and $z = 3$, arriving at the final answer of $11$. This journey through algebraic evaluation has highlighted several key concepts. We began by dissecting the expression, understanding the role of each term and the importance of fractional coefficients. We then mastered the art of substitution, replacing variables with their corresponding numerical values with precision. The order of operations, guided by PEMDAS, played a crucial role in performing the calculations accurately. Finally, we presented a step-by-step solution, solidifying our understanding and providing a clear roadmap for future problem-solving. The process of evaluating algebraic expressions is not merely a mechanical exercise; it's a fundamental skill that empowers us to solve real-world problems and make informed decisions. Whether we're calculating the trajectory of a projectile, designing a bridge, or analyzing financial data, the ability to manipulate and evaluate expressions is indispensable. By mastering this skill, we unlock a powerful tool for understanding and interacting with the world around us. So, let's continue to explore the fascinating world of algebra, building upon the foundations we've laid and embracing the challenges that lie ahead. With practice and perseverance, we can become proficient in algebraic manipulation and unlock the full potential of this powerful mathematical language.