Which Expression Is Equivalent To − 18 + 24 C -18 + 24c − 18 + 24 C ?A. − 6 ( 3 + 4 C -6(3 + 4c − 6 ( 3 + 4 C ] B. − 6 ( − 3 + 4 C -6(-3 + 4c − 6 ( − 3 + 4 C ] C. 6 ( − 3 + 4 C 6(-3 + 4c 6 ( − 3 + 4 C ] D. 6 ( 3 − 4 C 6(3 - 4c 6 ( 3 − 4 C ]

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 identifying equivalent expressions. We will use the given expression $-18 + 24c$ as a case study and examine the options provided to determine which one is equivalent.

Understanding the Given Expression

The given expression is $-18 + 24c$. This expression consists of two terms: a constant term $-18$ and a variable term $24c$. To simplify this expression, we need to understand the properties of algebraic expressions, including the distributive property and the concept of equivalent expressions.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses. In the context of the given expression, the distributive property can be used to rewrite the expression in a different form.

Option A: $-6(3 + 4c)$

Let's examine Option A: $-6(3 + 4c)$. To simplify this expression, we can use the distributive property to expand the expression:

$$-6(3 + 4c) = -6 \cdot 3 + (-6) \cdot 4c$$

Using the distributive property, we can rewrite the expression as:

$$-6(3 + 4c) = -18 - 24c$$

This expression is equivalent to the given expression $-18 + 24c$, but with a negative sign in front of the variable term.

Option B: $-6(-3 + 4c)$

Let's examine Option B: $-6(-3 + 4c)$. To simplify this expression, we can use the distributive property to expand the expression:

$$-6(-3 + 4c) = -6 \cdot (-3) + (-6) \cdot 4c$$

Using the distributive property, we can rewrite the expression as:

$$-6(-3 + 4c) = 18 - 24c$$

This expression is not equivalent to the given expression $-18 + 24c$.

Option C: $6(-3 + 4c)$

Let's examine Option C: $6(-3 + 4c)$. To simplify this expression, we can use the distributive property to expand the expression:

$$6(-3 + 4c) = 6 \cdot (-3) + 6 \cdot 4c$$

Using the distributive property, we can rewrite the expression as:

$$6(-3 + 4c) = -18 + 24c$$

This expression is equivalent to the given expression $-18 + 24c$.

Option D: $6(3 - 4c)$

Let's examine Option D: $6(3 - 4c)$. To simplify this expression, we can use the distributive property to expand the expression:

$$6(3 - 4c) = 6 \cdot 3 + 6 \cdot (-4c)$$

Using the distributive property, we can rewrite the expression as:

$$6(3 - 4c) = 18 - 24c$$

This expression is not equivalent to the given expression $-18 + 24c$.

Conclusion

In conclusion, the correct answer is Option C: $6(-3 + 4c)$. This expression is equivalent to the given expression $-18 + 24c$. The distributive property was used to simplify the expression and identify the equivalent expression.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses.
• To simplify an expression, we need to understand the properties of algebraic expressions, including the distributive property and the concept of equivalent expressions.
• The correct answer is Option C: $6(-3 + 4c)$, which is equivalent to the given expression $-18 + 24c$.

Final Answer

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Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses. It is denoted by the formula:

$$a(b + c) = ab + ac$$

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

A: To apply the distributive property, you need to multiply each term inside the parentheses by the factor outside the parentheses. For example, if you have the expression $a(b + c)$, you can simplify it by multiplying each term inside the parentheses by $a$:

$$a(b + c) = ab + ac$$

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as another expression. For example, the expressions $2x + 3$ and $x + 2x + 3$ are equivalent because they have the same value.

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

A: To determine if two expressions are equivalent, you need to simplify each expression and compare their values. If the values are the same, then the expressions are equivalent.

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

A: A variable is a letter or 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 variables and constants?

A: To simplify an expression with variables and constants, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, the expression $2x + 3x$ can be simplified by combining the like terms:

$$2x + 3x = 5x$$

Q: What is the order of operations in algebra?

A: The order of operations in algebra is:

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 apply the order of operations to simplify an expression?

A: To apply the order of operations, you need to follow the order of operations and evaluate each expression in the correct order. For example, if you have the expression $2x^2 + 3x - 4$, you would evaluate it as follows:

1. Parentheses: There are no expressions inside parentheses.
2. Exponents: Evaluate the exponential expression $x^2$.
3. Multiplication and Division: There are no multiplication or division operations.
4. Addition and Subtraction: Finally, evaluate the addition and subtraction operations from left to right.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with one variable and a degree of 1. A quadratic expression is an expression with one variable and a degree of 2.

Q: How do I simplify a linear expression?

A: To simplify a linear expression, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, the expression $2x + 3x$ can be simplified by combining the like terms:

$$2x + 3x = 5x$$

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, the expression $x^2 + 2x + 3$ can be simplified by combining the like terms:

$$x^2 + 2x + 3 = x^2 + 2x + 3$$

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics. By understanding the distributive property, equivalent expressions, and the order of operations, you can simplify complex expressions and solve problems with ease. Remember to combine like terms, apply the order of operations, and evaluate expressions in the correct order to simplify expressions.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses.
• To simplify an expression, you need to combine like terms, apply the order of operations, and evaluate expressions in the correct order.
• A linear expression is an expression with one variable and a degree of 1, while a quadratic expression is an expression with one variable and a degree of 2.

Final Answer

The final answer is that simplifying algebraic expressions is an essential skill in mathematics that requires understanding the distributive property, equivalent expressions, and the order of operations. By combining like terms, applying the order of operations, and evaluating expressions in the correct order, you can simplify complex expressions and solve problems with ease.