Which Expression Is Equivalent To $24x + 12$? Select All That Apply.A. $2(12x + 6)$ B. $ − 6 ( − 4 X − 2 ) -6(-4x - 2) − 6 ( − 4 X − 2 ) [/tex] C. $-3(8x - 4)$ D. $12(1 + 2x)$ E. $ − 4 ( − 6 − 3 X ) -4(-6 - 3x) − 4 ( − 6 − 3 X ) [/tex]
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore the concept of equivalent expressions and provide a step-by-step guide on how to simplify algebraic expressions. We will also examine the given options and determine which ones are equivalent to the expression $24x + 12$.
What are Equivalent Expressions?
Equivalent expressions are algebraic expressions that have the same value, but may be written in different forms. In other words, they are expressions that can be transformed into each other through a series of mathematical operations. For example, the expressions $2x + 3$ and $x + \frac{3}{2}$ are equivalent because they both represent the same value.
Simplifying Algebraic Expressions
To simplify an algebraic expression, we need to combine like terms and eliminate any unnecessary parentheses. Like terms are terms that have the same variable raised to the same power. For example, the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.
Step 1: Distribute the Coefficient
The first step in simplifying an algebraic expression is to distribute the coefficient to each term inside the parentheses. For example, if we have the expression $2(3x + 4)$, we can distribute the coefficient 2 to each term inside the parentheses to get $6x + 8$.
Step 2: Combine Like Terms
The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine these terms by adding their coefficients to get $5x$.
Step 3: Eliminate Unnecessary Parentheses
The final step is to eliminate any unnecessary parentheses. Parentheses are used to group terms together and indicate that they should be evaluated first. If we have an expression with parentheses, we can eliminate them by distributing the coefficient to each term inside the parentheses.
Analyzing the Options
Now that we have a basic understanding of how to simplify algebraic expressions, let's analyze the given options and determine which ones are equivalent to the expression $24x + 12$.
Option A: $2(12x + 6)$
To simplify this expression, we need to distribute the coefficient 2 to each term inside the parentheses. This gives us $24x + 12$, which is equivalent to the original expression.
Option B: $-6(-4x - 2)$
To simplify this expression, we need to distribute the coefficient -6 to each term inside the parentheses. This gives us $24x + 12$, which is equivalent to the original expression.
Option C: $-3(8x - 4)$
To simplify this expression, we need to distribute the coefficient -3 to each term inside the parentheses. This gives us $-24x + 12$, which is not equivalent to the original expression.
Option D: $12(1 + 2x)$
To simplify this expression, we need to distribute the coefficient 12 to each term inside the parentheses. This gives us $12 + 24x$, which is not equivalent to the original expression.
Option E: $-4(-6 - 3x)$
To simplify this expression, we need to distribute the coefficient -4 to each term inside the parentheses. This gives us $24x + 12$, which is equivalent to the original expression.
Conclusion
In conclusion, the options that are equivalent to the expression $24x + 12$ are:
- Option A: $2(12x + 6)$
- Option B: $-6(-4x - 2)$
- Option E: $-4(-6 - 3x)$
These expressions can be simplified to $24x + 12$ by distributing the coefficient to each term inside the parentheses and combining like terms. The other options are not equivalent to the original expression and can be eliminated.
Final Thoughts
Q: What is the difference between an algebraic expression and an equation?
A: An algebraic expression is a mathematical statement that contains variables and constants, but does not have an equal sign (=). An equation, on the other hand, is a mathematical statement that contains an equal sign and is used to solve for a variable.
Q: How do I simplify an algebraic expression with parentheses?
A: To simplify an algebraic expression with parentheses, you need to distribute the coefficient to each term inside the parentheses. This means multiplying the coefficient by each term inside the parentheses.
Q: What is the order of operations when simplifying algebraic expressions?
A: The order of operations when simplifying algebraic expressions is:
- Parentheses: Evaluate any expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I combine like terms in an algebraic expression?
A: To combine like terms in an algebraic expression, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x + 3x$, you can combine the like terms by adding the coefficients to get $5x$.
Q: What is the difference between a coefficient and a constant?
A: A coefficient is a number that is multiplied by a variable in an algebraic expression. A constant, on the other hand, is a number that is not multiplied by a variable.
Q: How do I simplify an algebraic expression with negative coefficients?
A: To simplify an algebraic expression with negative coefficients, you need to distribute the negative coefficient to each term inside the parentheses. This means multiplying the negative coefficient by each term inside the parentheses.
Q: Can I simplify an algebraic expression by canceling out like terms?
A: Yes, you can simplify an algebraic expression by canceling out like terms. This means subtracting the coefficients of the like terms to get a simpler expression.
Q: How do I know 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 I simplify an algebraic expression by using algebraic properties?
A: Yes, you can simplify an algebraic expression by using algebraic properties such as the distributive property, the commutative property, and the associative property.
Q: How do I use algebraic properties to simplify an expression?
A: To use algebraic properties to simplify an expression, you need to identify the property that applies to the expression and apply it to simplify the expression.
Q: What are some common algebraic properties that I can use to simplify expressions?
A: Some common algebraic properties that you can use to simplify expressions include:
- The distributive property: $a(b + c) = ab + ac$
- The commutative property: $a + b = b + a$
- The associative property: $(a + b) + c = a + (b + c)$
Conclusion
In conclusion, simplifying algebraic expressions is a crucial skill for students to master. By understanding the concepts and techniques outlined in this article, students can simplify complex expressions and solve problems with confidence. Remember to always follow the order of operations, combine like terms, and use algebraic properties to simplify expressions.