Which Expression Is Equivalent To 8 T − 18 − 20 T + 14 8t - 18 - 20t + 14 8 T − 18 − 20 T + 14 ?A. 3 ( 4 T − 9 − 10 T + 7 ) 2 ( 8 T − 18 − 20 T + 14 ) \frac{3(4t - 9 - 10t + 7)}{2(8t - 18 - 20t + 14)} 2 ( 8 T − 18 − 20 T + 14 ) 3 ( 4 T − 9 − 10 T + 7 ) B. 2 ( 4 T − 9 − 10 T + 7 ) 1 2 ( 8 T − 18 − 20 T + 14 ) \frac{2(4t - 9 - 10t + 7)}{\frac{1}{2}(8t - 18 - 20t + 14)} 2 1 ( 8 T − 18 − 20 T + 14 ) 2 ( 4 T − 9 − 10 T + 7 )

Mar 13, 2025 by ADMIN 430 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression $8t - 18 - 20t + 14$ . We will also examine two possible equivalent expressions and determine which one is correct.

Understanding the Given Expression

The given expression is $8t - 18 - 20t + 14$ . To simplify this expression, we need to combine like terms. Like terms are terms that have the same variable raised to the same power.

8t - 18 - 20t + 14

Step 1: Combine Like Terms

The first step in simplifying the expression is to combine the like terms. In this case, we have two terms with the variable $t$ : $8t$ and $-20t$ . We can combine these terms by adding their coefficients.

8t - 20t = -12t

Now, we can rewrite the expression with the combined like terms.

-12t - 18 + 14

Step 2: Simplify the Constants

Next, we need to simplify the constants in the expression. We can do this by combining the two constant terms: $-18$ and $14$ .

-18 + 14 = -4

Now, we can rewrite the expression with the simplified constants.

-12t - 4

Step 3: Write the Expression in a Different Form

The expression $-12t - 4$ is equivalent to the original expression $8t - 18 - 20t + 14$ . However, we can also write it in a different form by factoring out the common factor of $-4$ .

-4(3t + 1)

This is an equivalent expression to the original expression.

Evaluating the Options

Now, let's evaluate the two options given in the problem.

Option A

The first option is $\frac{3(4t - 9 - 10t + 7)}{2(8t - 18 - 20t + 14)}$ . To determine if this expression is equivalent to the original expression, we need to simplify it.

\frac{3(4t - 9 - 10t + 7)}{2(8t - 18 - 20t + 14)}

We can start by simplifying the numerator.

3(4t - 9 - 10t + 7) = 3(-6t - 2)

Now, we can rewrite the expression with the simplified numerator.

\frac{3(-6t - 2)}{2(8t - 18 - 20t + 14)}

Next, we can simplify the denominator.

2(8t - 18 - 20t + 14) = 2(-12t - 4)

Now, we can rewrite the expression with the simplified denominator.

\frac{3(-6t - 2)}{2(-12t - 4)}

We can simplify this expression further by canceling out the common factor of $-6$ in the numerator and denominator.

\frac{3(-6t - 2)}{2(-12t - 4)} = \frac{-3(6t + 2)}{-4(6t + 2)}

Now, we can cancel out the common factor of $(6t + 2)$ in the numerator and denominator.

\frac{-3(6t + 2)}{-4(6t + 2)} = \frac{-3}{-4}

This expression is equivalent to the original expression.

Option B

The second option is $\frac{2(4t - 9 - 10t + 7)}{\frac{1}{2}(8t - 18 - 20t + 14)}$ . To determine if this expression is equivalent to the original expression, we need to simplify it.

\frac{2(4t - 9 - 10t + 7)}{\frac{1}{2}(8t - 18 - 20t + 14)}

We can start by simplifying the numerator.

2(4t - 9 - 10t + 7) = 2(-6t - 2)

Now, we can rewrite the expression with the simplified numerator.

\frac{2(-6t - 2)}{\frac{1}{2}(8t - 18 - 20t + 14)}

Next, we can simplify the denominator.

\frac{1}{2}(8t - 18 - 20t + 14) = \frac{1}{2}(-12t - 4)

Now, we can rewrite the expression with the simplified denominator.

\frac{2(-6t - 2)}{\frac{1}{2}(-12t - 4)}

We can simplify this expression further by canceling out the common factor of $-6$ in the numerator and denominator.

\frac{2(-6t - 2)}{\frac{1}{2}(-12t - 4)} = \frac{-4(6t + 2)}{-6(6t + 2)}

Now, we can cancel out the common factor of $(6t + 2)$ in the numerator and denominator.

\frac{-4(6t + 2)}{-6(6t + 2)} = \frac{-4}{-6}

This expression is not equivalent to the original expression.

Conclusion

In conclusion, the expression $8t - 18 - 20t + 14$ is equivalent to $-12t - 4$ . We also evaluated two possible equivalent expressions and determined that option A is correct.

\frac{3(4t - 9 - 10t + 7)}{2(8t - 18 - 20t + 14)} = -\frac{3}{4}

This expression is equivalent to the original expression.

Final Answer

Introduction

Q&A Section

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What is the process of simplifying an algebraic expression?

A: The process of simplifying an algebraic expression involves combining like terms, which are terms that have the same variable raised to the same power. This can be done by adding or subtracting the coefficients of the like terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have two terms with the variable $t$ : $8t$ and $-20t$ , you can combine them by adding their coefficients: $8t - 20t = -12t$ .

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you need to combine like terms, which are terms that have the same variable raised to the same power. This can be done by adding or subtracting the coefficients of the like terms.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to make the expression easier to understand and work with. It also helps to identify patterns and relationships between variables and constants.

Q: Can you provide an example of simplifying an algebraic expression?

A: Yes, here is an example of simplifying an algebraic expression:

8t - 18 - 20t + 14

To simplify this expression, we need to combine like terms. We can start by combining the terms with the variable $t$ : $8t$ and $-20t$ . This gives us:

-12t - 18 + 14

Next, we can combine the constant terms: $-18$ and $14$ . This gives us:

-12t - 4

This is the simplified expression.

Q: How do I determine if an algebraic expression is equivalent to another expression?

A: To determine if an algebraic expression is equivalent to another expression, you need to simplify both expressions and compare them. If the simplified expressions are the same, then the original expressions are equivalent.

Q: Can you provide an example of determining if two algebraic expressions are equivalent?

A: Yes, here is an example of determining if two algebraic expressions are equivalent:

\frac{3(4t - 9 - 10t + 7)}{2(8t - 18 - 20t + 14)}

To determine if this expression is equivalent to the original expression $8t - 18 - 20t + 14$ , we need to simplify both expressions and compare them. We can start by simplifying the numerator:

3(4t - 9 - 10t + 7) = 3(-6t - 2)

Next, we can simplify the denominator:

2(8t - 18 - 20t + 14) = 2(-12t - 4)

Now, we can rewrite the expression with the simplified numerator and denominator:

\frac{3(-6t - 2)}{2(-12t - 4)}

We can simplify this expression further by canceling out the common factor of $-6$ in the numerator and denominator:

\frac{3(-6t - 2)}{2(-12t - 4)} = \frac{-3(6t + 2)}{-4(6t + 2)}

Now, we can cancel out the common factor of $(6t + 2)$ in the numerator and denominator:

\frac{-3(6t + 2)}{-4(6t + 2)} = \frac{-3}{-4}

This expression is equivalent to the original expression.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By combining like terms and simplifying expressions, we can make the expression easier to understand and work with. We also need to determine if two algebraic expressions are equivalent by simplifying both expressions and comparing them.

Final Answer

The final answer is $\boxed{-\frac{3}{4}}$ .