Simplifying Algebraic Expressions A Step-by-Step Guide

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Hey guys! Today, we're going to dive into simplifying a mathematical expression. This might seem daunting at first, but we'll break it down step-by-step to make it super clear. Our expression is: (7-4x+2(5x+8))/(7-4x-10x-16). We will learn how to simplify complex expressions by combining like terms and applying the distributive property, ultimately making them more manageable and easier to understand. Let’s get started!

Understanding the Expression

Before we jump into simplifying, let's take a good look at what we have. The expression consists of a fraction where both the numerator (the top part) and the denominator (the bottom part) involve variables and constants. Our main goal here is to simplify each part separately and then see if we can further reduce the fraction.

  • Numerator: 7 - 4x + 2(5x + 8)
  • Denominator: 7 - 4x - 10x - 16

See those parentheses in the numerator? That means we'll need to use the distributive property. Don't worry, it's not as scary as it sounds! The distributive property simply means multiplying the term outside the parentheses by each term inside the parentheses. Understanding the structure of the expression is key to simplifying it correctly. By identifying the components—the numerator, the denominator, terms with variables, and constants—we can develop a clear strategy for simplification. This initial assessment helps prevent errors and ensures we apply the correct operations in the right order.

The numerator of the expression, 7 - 4x + 2(5x + 8), presents a combination of constants, terms with variables, and a parenthetical expression. Simplifying the numerator involves first applying the distributive property to eliminate the parentheses and then combining like terms. This process not only reduces the complexity of the numerator but also makes it easier to compare and combine with other parts of the expression. Paying close attention to the signs of each term is crucial to avoid common arithmetic mistakes. For example, distributing the 2 across (5x + 8) requires careful multiplication to ensure both 5x and 8 are correctly accounted for. Once the distributive property is applied, identifying and combining the x terms and the constants will further simplify the numerator, setting the stage for the next steps in solving the overall expression.

Similarly, the denominator, 7 - 4x - 10x - 16, consists of constants and terms with the variable x. Simplifying the denominator involves combining like terms, which means grouping the x terms together and the constants together. This process streamlines the expression, making it easier to understand and work with. Accurate simplification of the denominator is essential as it forms the base for the entire fraction. Any errors in this part can propagate through the rest of the solution. Therefore, carefully adding or subtracting the coefficients of x and the constants ensures the denominator is correctly simplified. The resulting simplified denominator will then be used in conjunction with the simplified numerator to further reduce the overall expression, if possible. For example, the constants 7 and -16 need to be combined, and the terms -4x and -10x also need to be combined to fully simplify the denominator.

Simplifying the Numerator

Okay, let's tackle the numerator first: 7 - 4x + 2(5x + 8).

  1. Distribute: The first step is to distribute the 2 across the terms inside the parentheses. Remember, this means multiplying 2 by both 5x and 8.

    • 2 * 5x = 10x
    • 2 * 8 = 16

    So now our numerator looks like this: 7 - 4x + 10x + 16

  2. Combine Like Terms: Next, we combine terms that are similar. Like terms are terms that have the same variable raised to the same power (or are just constants).

    • Combine the 'x' terms: -4x + 10x = 6x
    • Combine the constants: 7 + 16 = 23

    Therefore, the simplified numerator is 6x + 23.

To simplify the numerator, 7 - 4x + 2(5x + 8), the first key step is applying the distributive property. This involves multiplying the 2 outside the parentheses by each term inside the parentheses, which means both 5x and 8. This process effectively eliminates the parentheses and transforms the expression into a more manageable form. Accuracy in this step is crucial because any mistake in distribution can lead to an incorrect final answer. The distributive property not only simplifies the expression but also makes it easier to combine like terms in the subsequent steps. For example, multiplying 2 by 5x yields 10x, and multiplying 2 by 8 yields 16. Thus, the expression becomes 7 - 4x + 10x + 16, setting the stage for combining the x terms and constants.

After applying the distributive property and obtaining the expression 7 - 4x + 10x + 16, the next crucial step is to combine like terms. Like terms are those that have the same variable raised to the same power (e.g., -4x and 10x) or are constants (e.g., 7 and 16). Combining like terms simplifies the expression by reducing the number of terms and making it easier to understand. This step involves adding or subtracting the coefficients of the variable terms and adding or subtracting the constants. Attention to the signs (positive or negative) of each term is essential to ensure correct combination. For instance, combining -4x and 10x requires subtracting 4x from 10x, resulting in 6x. Similarly, combining the constants 7 and 16 involves adding them together, yielding 23. Thus, the simplified numerator becomes 6x + 23, a more concise form that maintains the original mathematical relationship.

Simplifying the Denominator

Now, let's simplify the denominator: 7 - 4x - 10x - 16.

  1. Combine Like Terms: Here, we just need to combine like terms since there are no parentheses to deal with.

    • Combine the 'x' terms: -4x - 10x = -14x
    • Combine the constants: 7 - 16 = -9

    So, the simplified denominator is -14x - 9.

The denominator 7 - 4x - 10x - 16 is simplified by combining like terms, which include the x terms and the constant terms. This process involves adding or subtracting the coefficients of the x terms and the constants to reduce the expression to its simplest form. It is crucial to accurately identify and group the like terms, paying close attention to the signs preceding each term. Combining like terms not only simplifies the expression but also makes it easier to compare with other parts of the overall fraction. For example, combining -4x and -10x requires adding their coefficients, resulting in -14x. Similarly, combining the constants 7 and -16 involves subtracting 16 from 7, which yields -9. The simplified denominator is thus -14x - 9, representing a more concise form that maintains the original mathematical equivalence. This step is vital for further simplification of the entire expression.

The Simplified Expression

We've done the hard work! Now we have:

  • Simplified Numerator: 6x + 23
  • Simplified Denominator: -14x - 9

Putting it all together, our simplified expression is (6x + 23) / (-14x - 9).

At this point, we check if the numerator and denominator have any common factors that can be further cancelled out. In this case, 6x + 23 and -14x - 9 do not share any common factors, so we cannot simplify the fraction any further. The final simplified form of the expression is (6x + 23) / (-14x - 9).

After separately simplifying the numerator and the denominator, the final step is to combine these simplified forms back into a fraction and assess whether further simplification is possible. The numerator, simplified to 6x + 23, and the denominator, simplified to -14x - 9, are now placed together as a fraction: (6x + 23) / (-14x - 9). To determine if additional simplification is feasible, we look for common factors between the numerator and the denominator. If both the numerator and the denominator share a factor, dividing both by that factor will reduce the fraction to its simplest form. However, in this specific case, there are no common factors between 6x + 23 and -14x - 9. This lack of common factors means that the fraction is already in its simplest form. Therefore, the final simplified expression is (6x + 23) / (-14x - 9), representing the most concise form of the original expression.

Conclusion

And there you have it! We successfully simplified the expression (7-4x+2(5x+8))/(7-4x-10x-16) to (6x + 23) / (-14x - 9). Remember, the key is to break down the problem into smaller steps: distribute, combine like terms, and then see if you can simplify the fraction further. Keep practicing, and you'll be a pro at simplifying expressions in no time! Guys, I hope this explanation helped you understand the process better. Happy simplifying!

Through this step-by-step simplification, we’ve seen how complex expressions can be made manageable by applying basic algebraic principles. The process of simplifying fractions involves first addressing the numerator and the denominator separately. For the numerator, the distributive property plays a critical role when parentheses are involved, ensuring that each term inside the parentheses is correctly multiplied by the term outside. Once the parentheses are eliminated, the next step is to combine like terms. Like terms are those that contain the same variable raised to the same power or are constants. Combining these terms simplifies the expression by reducing the number of terms. The same principles apply to simplifying the denominator: identify and combine like terms to reduce its complexity. Finally, once both the numerator and the denominator are simplified, they are combined back into a single fraction. At this point, it’s essential to check if further simplification is possible by looking for common factors between the numerator and the denominator. If a common factor exists, dividing both by that factor will yield the simplest form of the expression. If no common factors are present, the expression is considered fully simplified. This methodical approach not only simplifies the expression but also enhances understanding of the underlying algebraic principles.