Swati Is Substituting T = 5 T = 5 T = 5 And T = 9 T = 9 T = 9 To Determine If The Two Expressions Are Equivalent:Expression 1: 8 ( 4 T − 3 8(4t - 3 8 ( 4 T − 3 ] Expression 2: 32 T − 24 32t - 24 32 T − 24 Which Statement Is True?A. Both Expressions Are Equivalent To 264 When

Understanding the Problem

In mathematics, equivalence of algebraic expressions is a crucial concept that helps us determine whether two expressions represent the same value or not. Swati is tasked with evaluating the equivalence of two expressions by substituting specific values of the variable $t$. In this article, we will explore the given expressions, substitute the values of $t$, and determine which statement is true.

The Given Expressions

Expression 1: $8(4t - 3)$ Expression 2: $32t - 24$

Substituting Values of $t$

Swati is substituting $t = 5$ and $t = 9$ to determine if the two expressions are equivalent. Let's evaluate each expression for these values of $t$.

Evaluating Expression 1 for $t = 5$

To evaluate Expression 1 for $t = 5$, we substitute $t = 5$ into the expression:

$8(4t - 3)$ $= 8(4(5) - 3)$ $= 8(20 - 3)$ $= 8(17)$ $= 136$

Evaluating Expression 1 for $t = 9$

To evaluate Expression 1 for $t = 9$, we substitute $t = 9$ into the expression:

$8(4t - 3)$ $= 8(4(9) - 3)$ $= 8(36 - 3)$ $= 8(33)$ $= 264$

Evaluating Expression 2 for $t = 5$

To evaluate Expression 2 for $t = 5$, we substitute $t = 5$ into the expression:

$32t - 24$ $= 32(5) - 24$ $= 160 - 24$ $= 136$

Evaluating Expression 2 for $t = 9$

To evaluate Expression 2 for $t = 9$, we substitute $t = 9$ into the expression:

$32t - 24$ $= 32(9) - 24$ $= 288 - 24$ $= 264$

Analyzing the Results

From the evaluations above, we can see that:

• Expression 1 is equal to 136 when $t = 5$.
• Expression 1 is equal to 264 when $t = 9$.
• Expression 2 is equal to 136 when $t = 5$.
• Expression 2 is equal to 264 when $t = 9$.

Conclusion

Based on the evaluations, we can conclude that:

• Both expressions are equivalent when $t = 9$.
• Expression 1 is not equivalent to Expression 2 when $t = 5$.

Therefore, the correct statement is:

A. Both expressions are equivalent to 264 when $t = 9$.

Final Thoughts

Understanding the Concept of Equivalence

In mathematics, equivalence of algebraic expressions is a crucial concept that helps us determine whether two expressions represent the same value or not. In this article, we will address some frequently asked questions (FAQs) related to the equivalence of algebraic expressions.

Q: What is the significance of equivalence in algebra?

A: Equivalence in algebra is significant because it helps us determine whether two expressions represent the same value or not. This concept is essential in solving equations, simplifying expressions, and evaluating the validity of mathematical statements.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can substitute specific values of the variable into both expressions and compare the results. If the results are the same, then the expressions are equivalent.

Q: What are some common methods for determining equivalence?

A: Some common methods for determining equivalence include:

• Substitution method: Substitute specific values of the variable into both expressions and compare the results.
• Factoring method: Factor both expressions and compare the factors.
• Simplifying method: Simplify both expressions and compare the simplified forms.

Q: Can two expressions be equivalent even if they look different?

A: Yes, two expressions can be equivalent even if they look different. For example, the expressions $2x + 3$ and $x + \frac{3}{2}$ are equivalent, even though they look different.

Q: How do I know if an expression is equivalent to a given expression?

A: To determine if an expression is equivalent to a given expression, you can use the methods mentioned above, such as substitution, factoring, or simplifying.

Q: Can I use a calculator to determine if two expressions are equivalent?

A: Yes, you can use a calculator to determine if two expressions are equivalent. However, it's essential to understand the concept of equivalence and how to apply it to different types of expressions.

Q: What are some real-world applications of equivalence in algebra?

A: Equivalence in algebra has numerous real-world applications, including:

• Solving equations in physics and engineering
• Simplifying expressions in computer programming
• Evaluating the validity of mathematical statements in finance and economics

Q: Can I use equivalence to solve equations with variables on both sides?

A: Yes, you can use equivalence to solve equations with variables on both sides. By applying the concept of equivalence, you can isolate the variable and solve for its value.

Conclusion

In conclusion, equivalence of algebraic expressions is a fundamental concept in mathematics that helps us determine whether two expressions represent the same value or not. By understanding the concept of equivalence and applying it to different types of expressions, you can solve equations, simplify expressions, and evaluate the validity of mathematical statements. We hope this article has provided valuable insights into the concept of equivalence in algebra.