Which Expression Is Equivalent To 2.5 + 4 ( 3 4 A − 5 ) − 9.5 ? 2.5+4\left(\frac{3}{4} A-5\right)-9.5? 2.5 + 4 ( 4 3 ​ A − 5 ) − 9.5 ? A. 6.5 + 3 4 A − 5 − 9.5 6.5+\frac{3}{4} A-5-9.5 6.5 + 4 3 ​ A − 5 − 9.5 B. 2.5 + 3 A − 20 − 9.5 2.5+3a-20-9.5 2.5 + 3 A − 20 − 9.5 C. 2.5 + 4 + 3 4 A − 14.5 2.5+4+\frac{3}{4} A-14.5 2.5 + 4 + 4 3 ​ A − 14.5 D. 2.5 + 3 A − 5 − 9.5 2.5+3a-5-9.5 2.5 + 3 A − 5 − 9.5

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression: $2.5+4\left(\frac{3}{4} a-5\right)-9.5$. We will break down the expression step by step, using the order of operations and basic algebraic properties to simplify it.

Understanding the Order of Operations

Before we dive into simplifying the expression, it's essential to understand the order of operations. The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Expression

Now that we have a solid understanding of the order of operations, let's simplify the given expression step by step.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is $\frac{3}{4} a-5$. We will evaluate this expression first.

import sympy as sp
a = sp.symbols('a')

expression_inside_parentheses = (3/4)*a - 5
print(expression_inside_parentheses)

Step 2: Multiply the Expression Inside the Parentheses by 4

Now that we have evaluated the expression inside the parentheses, we can multiply it by 4.

# Multiply the expression inside the parentheses by 4
multiplied_expression = 4 * expression_inside_parentheses
print(multiplied_expression)

Step 3: Add 2.5 to the Result

Next, we add 2.5 to the result.

# Add 2.5 to the result
result = 2.5 + multiplied_expression
print(result)

Step 4: Subtract 9.5 from the Result

Finally, we subtract 9.5 from the result.

# Subtract 9.5 from the result
final_result = result - 9.5
print(final_result)

Comparing the Final Result to the Answer Choices

Now that we have simplified the expression, let's compare the final result to the answer choices.

A. $6.5+\frac{3}{4} a-5-9.5$ B. $2.5+3a-20-9.5$ C. $2.5+4+\frac{3}{4} a-14.5$ D. $2.5+3a-5-9.5$

The final result is $2.5+3a-14.5$. Comparing this to the answer choices, we can see that the correct answer is:

B. $2.5+3a-20-9.5$

Conclusion

$$$$

Q: What is the order of operations?

A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Multiply and divide any terms from left to right.
3. Add and subtract any terms from left to right.

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

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

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

A: To simplify an expression with multiple variables, follow these steps:

1. Evaluate any expressions inside parentheses.
2. Multiply and divide any terms from left to right.
3. Add and subtract any terms from left to right.

Q: What is the distributive property?

A: The distributive property is a rule that states that a single term can be multiplied by multiple terms. For example:

a(b + c) = ab + ac

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, follow these steps:

1. Identify the term that is being multiplied by multiple terms.
2. Multiply the term by each of the multiple terms.
3. Combine the resulting terms.

Q: What is the commutative property?

A: The commutative property is a rule that states that the order of the terms in an expression does not change the result. For example:

a + b = b + a

Q: How do I use the commutative property to simplify an expression?

A: To use the commutative property to simplify an expression, follow these steps:

1. Identify the terms in the expression.
2. Rearrange the terms to make the expression easier to simplify.
3. Combine the resulting terms.

Q: What is the associative property?

A: The associative property is a rule that states that the order in which we perform operations does not change the result. For example:

(a + b) + c = a + (b + c)

Q: How do I use the associative property to simplify an expression?

A: To use the associative property to simplify an expression, follow these steps:

1. Identify the operations in the expression.
2. Rearrange the operations to make the expression easier to simplify.
3. Combine the resulting terms.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By following the order of operations and using basic algebraic properties, we can simplify complex expressions step by step. In this article, we answered frequently asked questions about simplifying algebraic expressions and provided examples of how to use the distributive property, commutative property, and associative property to simplify expressions.