Distributive Property How To Solve Equations

When tackling algebraic equations, understanding the underlying logical reasons behind each step is crucial for success. One common type of problem involves identifying the property that justifies a particular transformation. In this comprehensive guide, we'll delve into the process of solving the equation $3(2-3x) + 4x = x - 7$, focusing specifically on the distributive property and its role in simplifying such expressions. Our goal is to equip you with the knowledge and skills necessary to confidently identify the correct logical reason for each step in the equation-solving journey.

Decoding the Equation: $3(2-3x) + 4x = x - 7$

Let's begin by examining the equation at hand: $3(2-3x) + 4x = x - 7$. This equation involves a variable, $x$, and several arithmetic operations. To solve for $x$, we need to isolate it on one side of the equation. This requires a series of steps, each justified by a specific property of equality. The first hurdle we encounter is the expression $3(2-3x)$, which calls for simplification using the distributive property. This property is a cornerstone of algebra, allowing us to eliminate parentheses and rearrange terms, thereby paving the way for further simplification. Before diving deeper, it's important to understand the various properties of equality that govern equation manipulation. These properties—including the addition, subtraction, multiplication, and division properties—ensure that we maintain the balance of the equation while transforming it. The distributive property, however, stands out as a key tool for expanding expressions and combining like terms, making it an essential concept for solving a wide range of algebraic problems.

Understanding the Properties of Equality

Before we delve into the distributive property, let's take a moment to appreciate the fundamental properties of equality. These properties serve as the bedrock of algebraic manipulation, ensuring that we can transform equations while preserving their balance and solutions. Four primary properties govern our actions: the addition property of equality, the subtraction property of equality, the multiplication property of equality, and the division property of equality. The addition property of equality dictates that adding the same value to both sides of an equation maintains the equality. For example, if we have the equation $a = b$, adding $c$ to both sides yields $a + c = b + c$. This seemingly simple principle is surprisingly powerful, allowing us to isolate variables by strategically adding terms to both sides of an equation. Similarly, the subtraction property of equality allows us to subtract the same value from both sides of an equation without disrupting the balance. If $a = b$, then $a - c = b - c$. This property is the inverse of the addition property, and together they form the foundation for moving terms across the equals sign. The multiplication property of equality extends this concept to multiplication, stating that multiplying both sides of an equation by the same non-zero value preserves the equality. So, if $a = b$, then $ac = bc$, provided that $c$ is not zero. This property is invaluable for eliminating fractions and clearing coefficients in equations. Lastly, the division property of equality mirrors the multiplication property, allowing us to divide both sides of an equation by the same non-zero value. If $a = b$, then $a/c = b/c$, again with the caveat that $c$ cannot be zero. This property helps us isolate variables when they are multiplied by a coefficient. While these properties are essential, they don't directly address expressions enclosed in parentheses. That's where the distributive property comes into play, providing a mechanism for simplifying such expressions and preparing them for further manipulation using the properties of equality.

The Distributive Property: A Key to Unlocking Equations

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. In essence, it dictates how to multiply a single term by a group of terms enclosed within parentheses. The general form of the distributive property is: $a(b + c) = ab + ac$. This means that we multiply the term outside the parentheses, $a$, by each term inside the parentheses, $b$ and $c$, and then add the resulting products. This property is not just a mathematical trick; it reflects the underlying logic of multiplication and its interaction with addition. To truly grasp the distributive property, consider a visual analogy. Imagine you have 3 bags, each containing 2 apples and 3 oranges. To find the total number of fruits, you could count the fruits in each bag (2 apples + 3 oranges = 5 fruits) and then multiply by the number of bags (3 bags * 5 fruits/bag = 15 fruits). Alternatively, you could calculate the total number of apples (3 bags * 2 apples/bag = 6 apples) and the total number of oranges (3 bags * 3 oranges/bag = 9 oranges) separately and then add them together (6 apples + 9 oranges = 15 fruits). The distributive property formalizes this second approach, allowing us to distribute the multiplication across the addition. Applying the distributive property often simplifies equations significantly. By eliminating parentheses, we can combine like terms and rearrange the equation into a more manageable form. This is precisely what we need to do in our equation, $3(2-3x) + 4x = x - 7$, to progress towards a solution. Recognizing the distributive property as the next logical step is crucial for solving this and many other algebraic equations.

Applying the Distributive Property to Our Equation

Now, let's apply the distributive property to the equation $3(2-3x) + 4x = x - 7$. The key here is to focus on the term $3(2-3x)$. According to the distributive property, we need to multiply the 3 by each term inside the parentheses: 2 and -3x. So, $3 * 2 = 6$, and $3 * (-3x) = -9x$. Therefore, $3(2-3x)$ simplifies to $6 - 9x$. Replacing $3(2-3x)$ with $6 - 9x$ in the original equation, we get: $6 - 9x + 4x = x - 7$. Notice how the distributive property has effectively eliminated the parentheses, transforming the equation into a more approachable form. This is a common strategy in algebra: use the distributive property to expand expressions and create opportunities for further simplification. After applying the distributive property, the next step often involves combining like terms. In our equation, we have two terms involving $x$: -9x and +4x. These can be combined to give -5x. So, our equation now becomes: $6 - 5x = x - 7$. From here, we can use the properties of equality—such as the addition and subtraction properties—to isolate the variable $x$ and solve for its value. However, the crucial step that sets the stage for these subsequent operations is the application of the distributive property. It's the key that unlocks the equation and allows us to move forward in the solution process. Recognizing and applying the distributive property correctly is a fundamental skill in algebra, and it's essential for success in solving a wide range of equations and problems.

Identifying the Correct Logical Reason

In the context of the original question, we were asked to identify the correct logical reason for the next step in solving the equation $3(2-3x) + 4x = x - 7$. We've established that the next step involves applying the distributive property to the term $3(2-3x)$. This allows us to rewrite the equation as $6 - 9x + 4x = x - 7$, eliminating the parentheses and paving the way for further simplification. Now, let's consider the answer choices provided:

A. Multiplication Property of Equality B. Addition Property of Equality C. Distributive Property of Equality D. Subtraction Property of Equality

Based on our detailed analysis, it's clear that the correct answer is C. Distributive Property of Equality. The distributive property is precisely the principle we used to expand the expression $3(2-3x)$. The other options, while valid properties of equality, are not directly applicable at this stage. The multiplication, addition, and subtraction properties come into play later in the solution process, when we manipulate the equation to isolate the variable $x$. However, the initial and most crucial step involves the distributive property. It's important to note that the phrase "Distributive Property of Equality" is slightly unconventional. The distributive property itself is a property of arithmetic operations, while the properties of equality dictate how we can manipulate equations while maintaining their balance. However, in the context of this question and the answer choices provided, option C is the most accurate and appropriate. Recognizing the specific property that justifies each step in solving an equation is a crucial skill in algebra. It demonstrates a deep understanding of the underlying principles and allows for accurate and efficient problem-solving.

Mastering Equation Solving: Key Takeaways

Solving algebraic equations is a fundamental skill in mathematics, and mastering it requires a solid understanding of the underlying principles and properties. In this guide, we've focused on the equation $3(2-3x) + 4x = x - 7$, highlighting the crucial role of the distributive property in simplifying expressions and paving the way for a solution. Let's recap the key takeaways from our exploration:

1. Properties of Equality: The addition, subtraction, multiplication, and division properties of equality are the foundation of equation manipulation. They allow us to perform operations on both sides of an equation while maintaining its balance and solutions.
2. The Distributive Property: The distributive property is essential for simplifying expressions involving parentheses. It dictates how to multiply a single term by a group of terms within parentheses: $a(b + c) = ab + ac$. Applying the distributive property often simplifies equations significantly, making them easier to solve.
3. Step-by-Step Approach: Solving equations requires a systematic, step-by-step approach. Each step should be justified by a specific property or principle. In our example, the distributive property was the necessary first step to eliminate parentheses and combine like terms.
4. Identifying the Correct Logical Reason: Recognizing the logical reason behind each step is crucial for developing a deep understanding of algebra. In the context of our equation, the distributive property was the clear choice for the initial simplification.
5. Combining Like Terms: After applying the distributive property, the next step often involves combining like terms. This simplifies the equation further and prepares it for the application of the properties of equality.
6. Practice Makes Perfect: Like any mathematical skill, equation solving requires practice. The more you work through different types of equations, the more comfortable and confident you'll become in applying the distributive property and other algebraic principles.

By understanding and applying these principles, you'll be well-equipped to tackle a wide range of algebraic equations and problems. The distributive property is a powerful tool in your mathematical arsenal, and mastering its use will significantly enhance your problem-solving abilities.

In conclusion, identifying the correct logical reason for the next step in solving an equation is a critical skill in algebra. When faced with the equation $3(2-3x) + 4x = x - 7$, the distributive property emerges as the key to unlocking the solution. By applying this property, we can eliminate parentheses, simplify expressions, and pave the way for further algebraic manipulations. The distributive property, in conjunction with the properties of equality, forms a powerful framework for solving equations and tackling mathematical problems. Mastering these concepts not only enhances your algebraic skills but also fosters a deeper appreciation for the logical structure of mathematics. So, embrace the distributive property, practice its application, and watch your equation-solving abilities soar. Remember, each step in the solution process is a logical consequence of a specific property or principle, and understanding these connections is the key to mathematical success.