Simplifying Algebraic Expressions Using Properties Of Operations

Algebraic expressions, the bedrock of mathematical problem-solving, often appear daunting at first glance. However, by understanding and applying the properties of operations, we can transform complex expressions into simpler, more manageable forms. This article delves into the process of simplifying an algebraic expression, providing a step-by-step guide with detailed explanations. Let's embark on this mathematical journey, focusing on the expression: $5(x-4) + 3x - 9x + 7$.

Step 1 Rewriting Subtraction as Addition The Foundation of Simplification

The initial step in simplifying the algebraic expression $5(x-4) + 3x - 9x + 7$ involves rewriting all subtraction operations as the addition of negative numbers. This transformation is crucial because it allows us to seamlessly apply the commutative and associative properties of addition in subsequent steps.

Subtraction, in essence, is the addition of the additive inverse. The additive inverse of a number is the number that, when added to the original number, results in zero. For instance, the additive inverse of 5 is -5, and the additive inverse of -3 is 3. Recognizing this relationship between subtraction and addition is fundamental to algebraic manipulation. So, the core concept here is understanding that subtracting a number is the same as adding its negative counterpart. This is a critical foundation for simplifying algebraic expressions, as it allows us to treat all terms as additions, which opens the door to using properties like commutativity and associativity more freely.

When we apply this principle to our expression, we replace the subtraction of $9x$ with the addition of $-9x$. The subtraction within the parentheses, $(x - 4)$, is also rewritten as the addition of $-4$, resulting in $(x + (-4))$. This seemingly small change is a powerful maneuver because it unifies the operations within the expression, making it easier to rearrange and combine terms later on. By converting subtraction into the addition of a negative number, we set the stage for applying other properties of operations, like the commutative and associative properties of addition, more effectively. This initial step is not just about changing symbols; it’s about changing our perspective on the expression, viewing it as a sum of terms, which is a much more flexible structure to work with.

Therefore, after rewriting the subtraction operations as addition, our expression transforms from $5(x-4) + 3x - 9x + 7$ to $5(x + (-4)) + 3x + (-9x) + 7$. This seemingly simple step is a cornerstone of algebraic simplification, paving the way for subsequent operations and ultimately leading to a more concise and understandable expression. The transformation highlights a key algebraic principle: subtraction is fundamentally the addition of a negative number.

Step 2 Applying the Distributive Property Expanding and Simplifying

Now that we've rewritten subtraction as addition, the next crucial step is to apply the distributive property. This property is a fundamental tool in algebra that allows us to eliminate parentheses and further simplify expressions. The distributive property states that for any numbers a, b, and c, $a(b + c) = ab + ac$. In simpler terms, it means we can multiply a term outside the parentheses by each term inside the parentheses.

In our expression, $5(x + (-4)) + 3x + (-9x) + 7$, the term $5$ is multiplied by the binomial $(x + (-4))$. Applying the distributive property, we multiply $5$ by both $x$ and $-4$. This gives us $5 * x + 5 * (-4)$, which simplifies to $5x + (-20)$. Remember, multiplying a positive number by a negative number results in a negative product.

The distributive property is essential because it allows us to break down complex expressions into smaller, more manageable parts. By removing the parentheses, we free up the terms inside to be combined with other like terms in the expression. This is a critical step in the simplification process because it allows us to consolidate terms and reduce the overall complexity of the expression.

After applying the distributive property, our expression now looks like this: $5x + (-20) + 3x + (-9x) + 7$. Notice how the parentheses are gone, and we have a series of terms connected by addition. This form is much easier to work with because we can now rearrange and combine like terms without the constraints of the parentheses. The distributive property is not just a mechanical rule; it’s a strategic tool that allows us to expand expressions and make them more accessible for simplification. Understanding and applying it correctly is a key skill in algebra.

By carefully applying the distributive property, we've taken a significant step towards simplifying our expression. We've transformed a more complex structure into a series of terms that can be easily combined and manipulated. This step highlights the power of algebraic properties in unraveling mathematical expressions and making them more understandable.

Step 3 Combining Like Terms Simplifying the Expression

After applying the distributive property, we arrive at the expression $5x + (-20) + 3x + (-9x) + 7$. The next vital step in our simplification journey is to combine like terms. Like terms are those that contain the same variable raised to the same power. In our expression, $5x$, $3x$, and $-9x$ are like terms because they all contain the variable $x$ raised to the power of 1. Similarly, $-20$ and $7$ are like terms because they are both constants.

To combine like terms, we simply add their coefficients. The coefficient is the number that multiplies the variable. For the $x$ terms, we have $5x + 3x + (-9x)$. Adding the coefficients, we get $5 + 3 + (-9) = -1$. Therefore, $5x + 3x + (-9x)$ simplifies to $-1x$, which is commonly written as $-x$.

For the constant terms, we have $-20 + 7$. Adding these, we get $-13$. So, combining the constant terms results in $-13$. Combining like terms is a fundamental process in algebra because it reduces the number of terms in an expression, making it more concise and easier to understand. It's like organizing a room; by grouping similar items together, you create a sense of order and clarity.

This step is crucial because it consolidates the expression, bringing together similar elements to create a more streamlined form. It’s not just about making the expression shorter; it’s about revealing its underlying structure and making it easier to work with in future calculations or problem-solving scenarios.

By carefully identifying and combining like terms, we've transformed our expression into a simpler form. This process highlights the importance of recognizing patterns and similarities within an expression. It's a skill that is honed through practice and is essential for mastering algebraic manipulation. This streamlined form lays the groundwork for any further operations or applications of the expression.

Final Result The Simplified Algebraic Expression

Having meticulously followed each step, we've successfully simplified the original algebraic expression, $5(x-4) + 3x - 9x + 7$. We began by rewriting subtraction as the addition of negative numbers, then applied the distributive property to eliminate parentheses, and finally, we combined like terms to consolidate the expression. The culmination of these efforts brings us to the final, simplified form:

$-x - 13$

This concise expression is mathematically equivalent to the original but is much easier to understand and work with. It showcases the power of algebraic manipulation and the importance of understanding and applying the properties of operations. The simplified expression is not just an answer; it’s a clear and efficient representation of the original expression’s value.

The journey from the initial complex expression to this final simplified form illustrates the elegance and efficiency of algebraic techniques. Each step we took was a deliberate move, guided by mathematical principles, to transform the expression into its most basic and understandable form. This final result is a testament to the power of algebraic simplification. It’s a concise and clear representation that makes it easier to grasp the underlying relationship between variables and constants.

By mastering these steps, you gain a valuable skill in algebra that can be applied to a wide range of problems. Simplification is not just about finding the right answer; it’s about developing a deeper understanding of mathematical structures and relationships. This ability to break down and simplify expressions is a cornerstone of advanced mathematical thinking.

<mark>Properties of Operations</mark>
<mark>Algebraic Expression</mark>
<mark>Simplification</mark>
<mark>Distributive Property</mark>
<mark>Combining Like Terms</mark>