Equivalent Expressions: Simplifying 3(x-9)

Let's break down the expression $3(x-9)$ and figure out which of the provided options is equivalent. This involves using the distributive property, a fundamental concept in algebra.

Understanding the Distributive Property

The distributive property states that for any numbers a, b, and c, the following is true:

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

a( b - c ) = a b - a c

In simpler terms, when you have a number multiplying a quantity inside parentheses, you multiply the number by each term inside the parentheses. This property is crucial for simplifying expressions and solving equations. Understanding and applying the distributive property correctly is a cornerstone of algebraic manipulation, allowing us to rewrite expressions in equivalent forms that are often easier to work with or understand.

For example, if we have $2(x + 3)$, we distribute the 2 to both the x and the 3, resulting in $2*x + 2*3$, which simplifies to $2x + 6$. Similarly, if we have $5(y - 4)$, we distribute the 5 to both the y and the -4, resulting in $5*y - 5*4$, which simplifies to $5y - 20$. The distributive property is not limited to simple expressions; it can also be applied to more complex algebraic expressions involving multiple terms and variables. Mastery of this property is essential for success in algebra and beyond.

Applying the Distributive Property to $3(x-9)$

Now, let’s apply the distributive property to our given expression, $3(x-9)$. Here, 3 is multiplying the quantity $(x-9)$. So, we need to multiply 3 by both x and -9.

$3(x-9) = 3 * x - 3 * 9$

This simplifies to:

$3x - 27$

Now, let's look at the answer choices to see which one matches our simplified expression.

Analyzing the Answer Choices

We need to determine which of the following options is equivalent to $3x - 27$:

A. $3(x)+9$

This option simplifies to $3x + 9$. This is incorrect because we should have a subtraction, not an addition, and the constant term should be 27, not 9.

B. $3(x)+3(9)$

This option simplifies to $3x + 27$. This is also incorrect because, similar to option A, we should have subtraction, not addition. The distributive property requires us to subtract in this case.

C. $3(x)-9$

This option simplifies to $3x - 9$. This is incorrect because while it does have the correct subtraction operation, the constant term is incorrect. It should be 27 (3 multiplied by 9), not 9.

D. $3(x)-3(9)$

This option simplifies to $3x - 27$. This is correct because it accurately applies the distributive property, multiplying 3 by both x and -9, resulting in the correct terms and operations.

Therefore, the correct answer is D.

Why is the Distributive Property Important?

The distributive property is a cornerstone of algebra for several reasons:

• Simplifying Expressions: It allows us to rewrite expressions in a more manageable form, combining like terms and making them easier to understand.
• Solving Equations: It's essential for solving equations, especially those involving parentheses. By distributing, we can isolate variables and find their values.
• Factoring: The distributive property works in reverse for factoring. Identifying a common factor in an expression allows us to "undistribute" it, simplifying the expression and potentially revealing hidden relationships.
• Advanced Mathematics: The concepts behind the distributive property extend to more advanced areas of mathematics, like calculus and linear algebra.

Without a solid grasp of the distributive property, many algebraic manipulations become significantly more difficult, if not impossible. It's a foundational tool that unlocks a wide range of problem-solving techniques.

In essence, mastering the distributive property is akin to learning the alphabet in reading; it's a fundamental building block upon which more complex mathematical skills are built. Its correct application ensures accurate simplification and manipulation of algebraic expressions, which is critical for success in mathematics.

Common Mistakes to Avoid

When applying the distributive property, it's easy to make mistakes. Here are some common pitfalls to watch out for:

• Forgetting to Distribute to All Terms: Make sure you multiply the term outside the parentheses by every term inside. A common error is to only multiply by the first term and forget the others.
• Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. Remember that a negative times a negative is a positive, and a negative times a positive is a negative.
• Incorrect Multiplication: Double-check your multiplication. A simple arithmetic error can throw off the entire problem.
• Not Combining Like Terms: After distributing, simplify the expression by combining any like terms (terms with the same variable and exponent). For example, after distributing, you might have an expression like $2x + 3x - 5$. You should combine the $2x$ and $3x$ to get $5x - 5$.
• Misunderstanding the Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Distribute before you add, subtract, multiply, or divide (unless the operation is within the parentheses).

By being aware of these common mistakes and taking the time to carefully apply the distributive property, you can significantly improve your accuracy and confidence in algebra.

Conclusion

Therefore, the expression $3(x-9)$ is equivalent to $3(x) - 3(9)$, which corresponds to option D. Understanding and correctly applying the distributive property is essential for simplifying algebraic expressions and solving equations. By avoiding common mistakes and practicing regularly, you can master this fundamental concept.

For further learning on algebraic expressions and the distributive property, you can visit Khan Academy's Algebra Section.