Simplifying Expressions: Finding The Correct Intermediate Step

Let's dive into the world of algebraic expressions! In this article, we'll break down how to simplify expressions, focusing on the crucial intermediate steps. We'll use the example of Jaleesa, who's working with the expression 2(x+3) + 5x - 3(2x-1). We'll explore the steps she might take, paying close attention to the distributive property and combining like terms, to identify the correct intermediate equation.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra, guys. It's the key to unlocking expressions that have parentheses. Basically, it tells us how to multiply a single term by a group of terms inside parentheses. The distributive property is the cornerstone of simplifying expressions, and mastering it is crucial for success in algebra. To truly grasp its power, let's break down its mechanics and explore why it works the way it does. The distributive property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. In simpler terms, when you multiply a number (a) by a sum (b + c), you can distribute the multiplication across each term inside the parentheses. This means you multiply 'a' by 'b' and then add it to the product of 'a' and 'c'. Let's consider a concrete example to illustrate this: 3(2 + 4). Using the distributive property, we multiply 3 by both 2 and 4: 3 * 2 + 3 * 4 = 6 + 12 = 18. You can verify this by first adding the numbers inside the parentheses and then multiplying: 3(6) = 18. The distributive property applies equally well when dealing with variables. For instance, in the expression 2(x + 5), we distribute the 2 to both 'x' and '5': 2 * x + 2 * 5 = 2x + 10. This principle is essential for expanding and simplifying algebraic expressions. Now, let's delve into why the distributive property works. It's fundamentally based on the concept of repeated addition. Multiplication, after all, is a shortcut for adding the same number multiple times. Consider the expression 4(x + 2). This can be interpreted as adding the group (x + 2) four times: (x + 2) + (x + 2) + (x + 2) + (x + 2). If we rearrange the terms, we get: x + x + x + x + 2 + 2 + 2 + 2, which simplifies to 4x + 8. This is exactly what we would obtain by applying the distributive property directly: 4 * x + 4 * 2 = 4x + 8. The distributive property not only simplifies calculations but also provides a structured approach to handling complex expressions. It ensures that each term within the parentheses is correctly accounted for, leading to accurate results. In more advanced algebra, the distributive property is used extensively in factoring, expanding polynomials, and solving equations. Its understanding is therefore paramount for anyone looking to excel in mathematics. Whether you're a student just beginning to learn algebra or someone revisiting the basics, mastering the distributive property is a valuable investment. It's a tool that will serve you well in various mathematical contexts, making complex problems more manageable and fostering a deeper understanding of algebraic principles. So, embrace the distributive property, practice its application, and watch as it transforms your ability to simplify and solve mathematical expressions.

Applying the Distributive Property to Jaleesa's Expression

In Jaleesa's expression, 2(x+3) + 5x - 3(2x-1), we see the distributive property in action twice. First, we need to distribute the '2' across '(x+3)' and then the '-3' across '(2x-1)'. Remember, guys, the sign in front of the number is part of the term! To properly apply the distributive property to Jaleesa's expression, let's break down each step systematically. The expression is 2(x + 3) + 5x - 3(2x - 1). We'll focus on distributing the terms outside the parentheses to the terms inside. First, consider the term 2(x + 3). We distribute the '2' to both 'x' and '3':

• 2 * x = 2x
• 2 * 3 = 6

So, 2(x + 3) becomes 2x + 6. Next, we move on to the term -3(2x - 1). It's crucial to remember that we're distributing a '-3', not just '3'. This means we need to pay close attention to the signs:

• -3 * 2x = -6x
• -3 * -1 = +3

So, -3(2x - 1) becomes -6x + 3. Now, we can rewrite the original expression with the distributive property applied:

2x + 6 + 5x - 6x + 3

This step is a crucial intermediate point in simplifying the expression. It showcases the direct application of the distributive property, which is the foundation for further simplification. Let's take a moment to analyze why this step is so important. The distributive property allows us to eliminate the parentheses, which are essentially barriers to combining like terms. By expanding the expression, we make it possible to identify and group terms that share the same variable or are constants. Without this step, the expression would remain in a less manageable form. Moreover, the act of distributing terms correctly involves a careful understanding of multiplication and sign conventions. For instance, multiplying a negative number by a negative number results in a positive number, a rule that is frequently tested in algebraic manipulations. A mistake in applying the distributive property can lead to an incorrect simplification, ultimately affecting the final answer. This intermediate step also serves as a checkpoint. By carefully reviewing this stage, we can catch any errors in distribution before moving on to combining like terms. It's always a good practice to double-check each multiplication and ensure that the signs are correct. In essence, this step transforms the expression from a condensed form with parentheses to an expanded form that is ready for further simplification. It's a pivotal moment in the simplification process, where the structure of the expression is fundamentally altered. By mastering this step, you gain greater confidence in your ability to manipulate algebraic expressions. You're not just following a rule; you're actively reshaping the expression to make it easier to work with. This active engagement with the material is what truly solidifies understanding and builds mathematical fluency.

Combining Like Terms

Once we've applied the distributive property, the next step is to combine like terms. Like terms are those that have the same variable raised to the same power (e.g., 2x and 5x) or are constants (e.g., 6 and 3). We can add or subtract like terms, but we can't combine terms that are not alike (e.g., we can't combine 2x and 6). Combining like terms is a fundamental step in simplifying algebraic expressions, and it's essential for reducing complexity and making equations easier to solve. The process involves identifying terms that share the same variable and exponent, or terms that are constants, and then adding or subtracting their coefficients. To truly master this skill, let's delve into the mechanics of combining like terms and understand why it works the way it does. Consider the expression 3x + 2y + 5x - y + 4. Our goal is to identify and combine the like terms. First, let's look at the terms involving the variable 'x': 3x and 5x. These are like terms because they both have 'x' raised to the power of 1. To combine them, we simply add their coefficients: 3 + 5 = 8. So, 3x + 5x becomes 8x. Next, let's examine the terms involving the variable 'y': 2y and -y. These are also like terms. Remember that '-y' is the same as '-1y'. To combine them, we add their coefficients: 2 + (-1) = 1. So, 2y - y becomes 1y, which we typically write as just y. Finally, we have the constant term, 4. Since there are no other constants in the expression, we simply carry it over. Putting it all together, the simplified expression is 8x + y + 4. This process of combining like terms is rooted in the distributive property, which we discussed earlier. When we combine 3x + 5x to get 8x, we're essentially factoring out the common variable 'x': 3x + 5x = (3 + 5)x = 8x. This highlights the connection between the distributive property and combining like terms. Let's consider another example: 7a² - 4a + 2a² + 6a - 3. In this case, we have terms with 'a²' and terms with 'a'. We combine the 'a²' terms: 7a² + 2a² = 9a². We combine the 'a' terms: -4a + 6a = 2a. And we have the constant term, -3, which remains unchanged. The simplified expression is 9a² + 2a - 3. Combining like terms is not just about simplifying expressions; it's also about making them more manageable. Simplified expressions are easier to understand, evaluate, and use in further calculations. For instance, when solving equations, combining like terms is often a crucial step in isolating the variable and finding the solution. In more advanced algebra, combining like terms is used in various contexts, such as polynomial arithmetic, calculus, and linear algebra. The ability to quickly and accurately combine like terms is a valuable asset in any mathematical endeavor. Whether you're a student working on homework problems or a professional applying mathematical principles in your field, mastering this skill will make your work more efficient and effective. So, practice identifying and combining like terms in various expressions. Pay attention to the signs and the coefficients, and remember the underlying principle of factoring out the common variable. With practice, you'll become adept at simplifying expressions and unlocking their underlying structure.

Combining Like Terms in Jaleesa's Expression

Looking back at Jaleesa's expression after applying the distributive property (2x + 6 + 5x - 6x + 3), we can identify the like terms: 2x, 5x, and -6x are like terms because they all contain the variable 'x'. The constants 6 and 3 are also like terms. Let's combine them: First, we combine the 'x' terms: 2x + 5x - 6x = (2 + 5 - 6)x = 1x, which we can simply write as x. Next, we combine the constants: 6 + 3 = 9. So, after combining like terms, Jaleesa's expression simplifies to x + 9. This step of combining like terms is a critical juncture in simplifying algebraic expressions, serving as a bridge between the expanded form obtained from the distributive property and the most concise, simplified form. The process involves bringing together terms that share the same variable and exponent, or constant terms, to reduce the expression to its simplest components. To fully appreciate the significance of this step, let's delve into its underlying principles and practical applications. When we combine like terms, we're essentially streamlining the expression by grouping together elements that can be treated as a single unit. This not only makes the expression more visually appealing but also more manageable for further calculations or manipulations. Consider the expression 4y + 7 - 2y + 3 - y. To combine like terms, we first identify the terms involving the variable 'y': 4y, -2y, and -y. We then add their coefficients: 4 + (-2) + (-1) = 1. So, 4y - 2y - y simplifies to y. Next, we combine the constant terms: 7 + 3 = 10. The simplified expression is y + 10. This process is grounded in the fundamental properties of addition and subtraction. We can rearrange the terms in an expression without changing its value, thanks to the commutative property of addition. This allows us to group like terms together, making it easier to combine them. For instance, in the expression 3a + 5b - a + 2b, we can rearrange the terms to group the 'a' terms and the 'b' terms: 3a - a + 5b + 2b. Then, we combine the 'a' terms: 3a - a = 2a, and the 'b' terms: 5b + 2b = 7b. The simplified expression is 2a + 7b. The importance of combining like terms extends beyond mere simplification. It's a crucial step in solving equations and inequalities. By reducing the number of terms in an equation, we make it easier to isolate the variable and find the solution. For example, if we have the equation 2x + 5 - x + 3 = 10, we first combine like terms on the left side: 2x - x = x and 5 + 3 = 8. The equation becomes x + 8 = 10. Now, it's much easier to solve for 'x' by subtracting 8 from both sides: x = 2. Combining like terms is also essential in working with polynomials. Polynomials are algebraic expressions consisting of variables and coefficients, such as 3x² - 2x + 1. When adding or subtracting polynomials, we combine like terms to simplify the result. For instance, if we add the polynomials (2x² + 3x - 4) and (x² - 5x + 2), we combine the 'x²' terms: 2x² + x² = 3x², the 'x' terms: 3x - 5x = -2x, and the constant terms: -4 + 2 = -2. The resulting polynomial is 3x² - 2x - 2. Mastering the art of combining like terms is a fundamental skill in algebra and beyond. It's a tool that simplifies complex expressions, streamlines calculations, and makes mathematical problem-solving more efficient. So, embrace the process, practice its application, and watch as your algebraic skills flourish.

Identifying the Correct Intermediate Step

Now, let's circle back to the original question. We're looking for a correct intermediate step in simplifying 2(x+3) + 5x - 3(2x-1). We've already determined that after applying the distributive property, the expression becomes 2x + 6 + 5x - 6x + 3. This is a key intermediate step. Comparing this to the options provided, we can identify the correct one. Remember, an intermediate step is a step along the way, not necessarily the final answer. Therefore, the correct intermediate step showcases the expression after the distributive property has been applied but before like terms have been fully combined. Let's analyze why this particular step is so crucial in the simplification process. After applying the distributive property, we've essentially