Evaluate The Expressions E. 3t(t-2), F. 3t(2-t), And G. (t-2)3t By Substituting T=5. Which Expressions Yield The Same Result?

Introduction

In this article, we will delve into the world of algebraic expressions and explore how to evaluate them by substituting a given value for a variable. Specifically, we will be working with the expressions E. $3t(t-2)$, F. $3t(2-t)$, and G. $(t-2)3t$. Our primary goal is to substitute $t=5$ into each of these expressions, calculate their respective values, and then determine which expressions yield the same result. This exercise is fundamental in understanding how algebraic expressions behave and how different forms of the same expression can produce identical values under certain conditions. This exploration not only reinforces the basic principles of algebra but also highlights the importance of order of operations and the distributive property. Mastering these concepts is crucial for tackling more complex algebraic problems in the future.

1.3.1 Evaluating the Expressions

Let's begin by substituting $t=5$ into each of the given expressions and calculating the resulting values. This process involves replacing every instance of the variable $t$ with the numerical value 5 and then simplifying the expression according to the order of operations (PEMDAS/BODMAS), which prioritizes parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). By meticulously following this order, we can accurately determine the value of each expression for the given value of $t$. This step-by-step evaluation is essential for avoiding common errors and ensuring a clear understanding of the algebraic manipulation involved. Furthermore, this exercise provides a practical application of the fundamental principles of algebra, demonstrating how variables and expressions are used to represent and solve mathematical problems.

Expression E: $3t(t-2)$

To evaluate expression E, we substitute $t=5$ into the expression $3t(t-2)$. This gives us:

$3(5)(5-2)$

First, we simplify the expression within the parentheses:

$5-2 = 3$

Now, we substitute this back into the expression:

$3(5)(3)$

Next, we perform the multiplication from left to right:

$3 * 5 = 15$

$15 * 3 = 45$

Therefore, when $t=5$, the value of expression E is 45.

Expression F: $3t(2-t)$

Now, let's evaluate expression F by substituting $t=5$ into the expression $3t(2-t)$. This results in:

$3(5)(2-5)$

We begin by simplifying the expression inside the parentheses:

$2-5 = -3$

Substituting this back into the expression, we get:

$3(5)(-3)$

Performing the multiplication from left to right:

$3 * 5 = 15$

$15 * -3 = -45$

Thus, when $t=5$, the value of expression F is -45.

Expression G: $(t-2)3t$

Finally, we evaluate expression G by substituting $t=5$ into the expression $(t-2)3t$. This gives us:

$(5-2)3(5)$

First, we simplify the expression within the parentheses:

$5-2 = 3$

Substituting this back into the expression, we have:

$(3)3(5)$

Performing the multiplication from left to right:

$3 * 3 = 9$

$9 * 5 = 45$

Therefore, when $t=5$, the value of expression G is 45.

1.3.2 Identifying Equivalent Expressions

Having evaluated expressions E, F, and G for $t=5$, we can now compare the results to determine which expressions yield the same value. This comparison is a crucial step in understanding the equivalence of different algebraic forms. It demonstrates that while expressions may appear different on the surface, they can produce identical results for specific values of the variable. This concept is fundamental in algebra and is used extensively in simplifying expressions, solving equations, and understanding the relationships between different mathematical representations.

Comparing the Results

From our calculations, we found the following values:

• E: $3t(t-2) = 45$
• F: $3t(2-t) = -45$
• G: $(t-2)3t = 45$

By comparing these values, we observe that expressions E and G both resulted in a value of 45 when $t=5$. Expression F, on the other hand, resulted in a value of -45. This difference highlights the impact of the order of operations and the signs within the expressions. While E and G might look different initially, the distributive property and the commutative property of multiplication demonstrate their equivalence.

Conclusion on Equivalent Expressions

Therefore, we can conclude that expressions E ($3t(t-2)$) and G ($(t-2)3t$) give the same answer when $t=5$. This equivalence is a result of the commutative property of multiplication, which states that the order in which numbers are multiplied does not affect the product. In this case, multiplying $3t$ by $(t-2)$ yields the same result as multiplying $(t-2)$ by $3t$. Understanding and recognizing such equivalences is a critical skill in algebra, as it allows for the simplification of expressions and the efficient solving of equations. This exercise underscores the importance of not only evaluating expressions but also analyzing their structure and properties to identify relationships and equivalences.

Further Discussion on Algebraic Expressions

The Significance of Order of Operations

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), is a fundamental principle in mathematics. It dictates the sequence in which operations must be performed to ensure a consistent and accurate evaluation of mathematical expressions. In the context of algebraic expressions, adhering to the order of operations is crucial for correctly substituting values and simplifying the expression. The exercise of evaluating expressions E, F, and G for $t=5$ vividly illustrates this significance. For instance, the difference in the results between expressions F and the others arises directly from the subtraction within the parentheses and the subsequent multiplication. Had the order of operations not been followed meticulously, the results would have been erroneous, leading to a misunderstanding of the expression's behavior. Therefore, a thorough understanding and application of the order of operations is indispensable for anyone working with algebraic expressions.

The Distributive Property and Its Implications

The distributive property is a cornerstone of algebra, allowing us to simplify expressions involving multiplication over addition or subtraction. This property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$. In our exploration of expressions E, F, and G, the distributive property plays a key role in understanding their structure and equivalence. While we evaluated the expressions by direct substitution, recognizing the potential for distribution provides deeper insight. For example, in expression E ($3t(t-2)$), we could distribute the $3t$ across the terms inside the parentheses to obtain $3t^2 - 6t$. This equivalent form can be useful in various algebraic manipulations, such as solving equations or graphing functions. Similarly, understanding the distributive property helps in recognizing that expression G ($(t-2)3t$) is equivalent to E due to the commutative property of multiplication. However, expression F ($3t(2-t)$) yields a different result because the subtraction within the parentheses is in the reverse order, leading to a different sign when distributed. The distributive property, therefore, is not just a tool for simplification but also a lens through which we can analyze and understand the relationships between algebraic expressions.

The Role of Variables in Algebraic Expressions

Variables are the building blocks of algebraic expressions, serving as placeholders for unknown or changing values. The power of algebra lies in its ability to represent general relationships and solve problems in a way that is independent of specific numerical values. In the expressions we examined, the variable $t$ represents a number that can take on different values. By substituting $t=5$, we were able to evaluate the expressions for a specific instance, but the expressions themselves hold true for any value of $t$. This generality is what makes algebra so powerful. Understanding the role of variables is essential for interpreting and manipulating algebraic expressions. Variables allow us to express mathematical relationships concisely and to solve problems that would be difficult or impossible to address using only arithmetic. For example, we could use the expressions to explore how the value of each expression changes as $t$ varies, which is a fundamental concept in calculus and other advanced mathematical fields. Thus, the variable is not just a symbol but a key to unlocking the power of algebraic reasoning.

Practical Applications of Evaluating Expressions

The ability to evaluate algebraic expressions is not just an abstract mathematical skill; it has numerous practical applications in various fields. From physics and engineering to economics and computer science, the evaluation of expressions is a fundamental step in solving real-world problems. In physics, for example, formulas often involve variables that represent physical quantities, and evaluating these formulas for specific values allows us to make predictions or analyze experimental data. In engineering, expressions are used to model the behavior of systems, and evaluating these expressions helps engineers design and optimize structures and devices. In economics, algebraic models are used to represent economic relationships, and evaluating these models allows economists to forecast economic trends or assess the impact of policy changes. Even in everyday life, we encounter situations where evaluating expressions can be helpful, such as calculating the cost of a purchase with a discount or determining the amount of interest earned on an investment. The exercise of substituting values into expressions and simplifying them is, therefore, a valuable skill that transcends the classroom and has broad relevance in the modern world.

Conclusion

In this article, we have explored the evaluation of algebraic expressions by substituting a specific value for a variable. We meticulously evaluated expressions E, F, and G for $t=5$, demonstrating the importance of the order of operations and the distributive property. We also identified that expressions E and G yield the same result, highlighting the concept of equivalent expressions. This exercise underscores the fundamental principles of algebra and their application in solving mathematical problems. Furthermore, we discussed the broader significance of algebraic expressions and their practical applications in various fields, emphasizing the importance of mastering these concepts for future mathematical endeavors.