Which Expression Is Equivalent To The Given Expression?$\[ 4(a - 3) \\]A. \[$-8a\$\]B. \[$4a - 12\$\]C. \[$4a - 3\$\]D. \[$-4(a + 3)\$\]

Understanding the Problem

In this problem, we are given an algebraic expression and asked to find an equivalent expression from a set of options. The given expression is $4(a - 3)$, and we need to determine which of the options is equivalent to it.

Option A: $-8a$

Let's start by analyzing Option A, which is $-8a$. To determine if this option is equivalent to the given expression, we need to simplify the given expression and compare it with Option A.

The given expression is $4(a - 3)$. Using the distributive property, we can expand this expression as follows:

$4(a - 3) = 4a - 12$

Now, let's compare this expression with Option A, which is $-8a$. We can see that the two expressions are not equivalent, as the given expression has a constant term of $-12$, while Option A has no constant term.

Option B: $4a - 12$

Next, let's analyze Option B, which is $4a - 12$. We can see that this option is similar to the given expression, which is $4(a - 3)$. In fact, the given expression can be simplified to $4a - 12$ using the distributive property.

Therefore, we can conclude that Option B is equivalent to the given expression.

Option C: $4a - 3$

Now, let's analyze Option C, which is $4a - 3$. We can see that this option is not equivalent to the given expression, as the given expression has a constant term of $-12$, while Option C has a constant term of $-3$.

Option D: $-4(a + 3)$

Finally, let's analyze Option D, which is $-4(a + 3)$. We can simplify this expression using the distributive property as follows:

$-4(a + 3) = -4a - 12$

We can see that this expression is not equivalent to the given expression, as the given expression has a coefficient of $4$, while Option D has a coefficient of $-4$.

Conclusion

In conclusion, we have analyzed all the options and determined that Option B, which is $4a - 12$, is equivalent to the given expression.

Key Takeaways

• The distributive property can be used to expand and simplify algebraic expressions.
• When comparing two expressions, we need to consider both the coefficients and the constant terms.
• Equivalent expressions have the same value, but may have different forms.

Practice Problems

1. Simplify the expression $3(2x + 5)$ using the distributive property.
2. Determine if the expression $-2(3x - 4)$ is equivalent to the expression $-6x + 8$.
3. Simplify the expression $2(x + 3)$ using the distributive property.

Answer Key

1. $6x + 15$
2. Yes
3. $2x + 6$

Additional Resources

• Khan Academy: Algebra
• Mathway: Algebra Calculator
• Wolfram Alpha: Algebra Solver
  Frequently Asked Questions (FAQs) =====================================

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to expand and simplify algebraic expressions by multiplying a single term to multiple terms inside parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the single term outside the parentheses to each term inside the parentheses. For example, if you have the expression $3(2x + 5)$, you would multiply $3$ to each term inside the parentheses, resulting in $6x + 15$.

Q: What is the difference between equivalent expressions?

A: Equivalent expressions are expressions that have the same value, but may have different forms. For example, the expressions $4a - 12$ and $4(a - 3)$ are equivalent because they both simplify to 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 are some common mistakes to avoid when working with algebraic expressions?

A: Some common mistakes to avoid when working with algebraic expressions include:

• Forgetting to distribute a term to all terms inside parentheses
• Not simplifying expressions fully
• Not checking for equivalent expressions

Q: How can I practice working with algebraic expressions?

A: You can practice working with algebraic expressions by:

• Solving practice problems
• Using online resources such as Khan Academy or Mathway
• Working with a tutor or teacher

Q: What are some real-world applications of algebraic expressions?

A: Algebraic expressions have many real-world applications, including:

• Science and engineering: Algebraic expressions are used to model and solve problems in physics, chemistry, and other sciences.
• Economics: Algebraic expressions are used to model and analyze economic systems.
• Computer science: Algebraic expressions are used in computer programming and algorithm design.

Q: How can I use algebraic expressions to solve problems?

A: To use algebraic expressions to solve problems, you need to:

• Identify the problem and the variables involved
• Write an algebraic expression to represent the problem
• Simplify the expression and solve for the variable

Q: What are some common algebraic expressions?

A: Some common algebraic expressions include:

• Linear expressions: $ax + b$
• Quadratic expressions: $ax^2 + bx + c$
• Polynomial expressions: $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

Q: How can I simplify algebraic expressions?

A: To simplify algebraic expressions, you need to:

• Combine like terms
• Use the distributive property to expand expressions
• Cancel out common factors

Q: What are some tips for working with algebraic expressions?

A: Some tips for working with algebraic expressions include:

• Read the problem carefully and identify the variables involved
• Write an algebraic expression to represent the problem
• Simplify the expression and solve for the variable
• Check your work and make sure the expression is equivalent to the original problem.