Solving (18x=30):6=90-67 With Distributive Property
Hey guys! Today, we're going to dive deep into solving the equation (18x = 30) : 6 = 90 - 67 using the distributive property. This might seem a bit tricky at first, but don't worry, we'll break it down step by step so that you can master it. Understanding how to apply the distributive property is crucial in algebra, as it helps simplify complex equations and makes them easier to solve. We'll not only solve this specific equation but also discuss the underlying principles and how you can apply them to other problems. Let's get started and make math fun and understandable!
Understanding the Distributive Property
Before we jump into solving the equation, let's make sure we're all on the same page about the distributive property. In simple terms, the distributive property states that multiplying a sum or difference by a number is the same as multiplying each term in the sum or difference by that number individually. Mathematically, it’s expressed as a(b + c) = ab + ac or a(b - c) = ab - ac. This property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations.
Think of it like this: you have a group of items (b + c) and you want to multiply the entire group by a certain factor (a). Instead of adding the items first and then multiplying, the distributive property allows you to multiply each item individually by the factor and then add the results. This can be particularly useful when dealing with expressions involving variables, as it helps to eliminate parentheses and simplify the equation. For example, if you have 3(x + 2), you can distribute the 3 to both x and 2, resulting in 3x + 6. This makes the expression much easier to work with.
The distributive property also applies to subtraction. For instance, if you have 4(y - 3), you distribute the 4 to both y and -3, resulting in 4y - 12. The key is to ensure that the factor outside the parentheses is multiplied by each term inside the parentheses, paying close attention to the signs. Mastering this property is essential for success in algebra and will help you tackle more complex problems with confidence. So, remember, the distributive property is your friend when it comes to simplifying expressions and solving equations. Now that we have a solid understanding of the distributive property, let's move on to applying it to our equation.
Breaking Down the Equation (18x = 30) : 6 = 90 - 67
Now, let's tackle our equation: (18x = 30) : 6 = 90 - 67. At first glance, it might seem a bit intimidating, but don't worry, we'll break it down into manageable steps. The first thing we need to do is simplify both sides of the equation. On the right side, we have a simple subtraction: 90 - 67. This is straightforward and will give us a single number. On the left side, we have (18x = 30) : 6, which involves both multiplication and division. We need to address these operations carefully to isolate the variable x.
The left side of the equation, (18x = 30) : 6, is where we'll focus most of our attention initially. Remember, the goal is to isolate x, but we can't do that directly until we simplify the expression. The notation (18x = 30) : 6 might be a bit confusing, but it essentially means that the entire expression 18x = 30 is being divided by 6. We need to consider the order of operations (PEMDAS/BODMAS) to ensure we solve it correctly. However, before diving into division, let’s first acknowledge that this initial form of the equation presents a slight notational ambiguity. The more standard way to express this would involve either parentheses to clarify the order of operations or rewriting it as a fraction. For the purpose of this explanation, we'll interpret it in a way that allows us to apply the distributive property, but it’s important to recognize that the original notation could be clearer.
To clarify, we'll reframe the left side as if we're distributing the division by 6 across an expression. This might not be the direct application of the distributive property as commonly seen with multiplication, but it allows us to conceptually approach the problem in a similar way. We'll see how this unfolds as we proceed. By breaking down the equation into smaller parts and addressing each operation systematically, we'll be able to simplify it and eventually solve for x. So, let's take it one step at a time and make sure we understand each step before moving on.
Step-by-Step Solution
Okay, let's walk through the solution step-by-step to make sure everything is crystal clear. First, we'll rewrite the equation to clarify the operations. The original equation (18x = 30) : 6 = 90 - 67 can be interpreted as (18x/6) = (30/6) = 90 - 67. This interpretation allows us to see how the division by 6 might conceptually be "distributed," even though it's not a direct application of the distributive property in its traditional sense. This is a crucial step to ensure we're on the right track.
Now, let’s simplify both sides. On the left side, we have 18x/6 and 30/6. We can simplify these by performing the division. 18x/6 simplifies to 3x, and 30/6 simplifies to 5. On the right side, we have 90 - 67, which equals 23. So, our equation now looks like this: 3x = 5 = 23. It's important to note that the equation 3x = 5 = 23 seems to have an error, as it implies that 3x equals both 5 and 23 simultaneously, which is not possible. To correct this, let's backtrack and reconsider the initial interpretation of the equation.
Given the original equation (18x = 30) : 6 = 90 - 67, it’s more likely that the division by 6 applies only to the expression 18x = 30, and the entire result of that operation should then be equated to 90 - 67. Thus, a more accurate way to rewrite the equation would be to first solve 18x = 30 and then divide the result by 6, comparing it to the simplified right side. Let’s correct our approach accordingly. First, we solve 18x = 30 for x. To do this, we divide both sides by 18: x = 30/18. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. So, x = (30 ÷ 6) / (18 ÷ 6) = 5/3. Now we have the value of x.
Next, we substitute x = 5/3 back into the left side of the original equation and divide the expression by 6: (18 * (5/3)) : 6. First, we multiply 18 * (5/3). This equals (18 * 5) / 3 = 90/3 = 30. Now we divide this result by 6: 30 : 6 = 5. So, the left side of the equation simplifies to 5. On the right side, we still have 90 - 67, which equals 23. Therefore, the equation becomes 5 = 23, which is clearly not true. This indicates that there might be an issue with the original equation or its interpretation. However, we’ve systematically worked through the steps, addressing the potential for distributive property conceptually and solving for x in the context of the given operations.
Addressing Potential Errors and Clarifications
At this point, it's crucial to address the potential errors or ambiguities in the original equation, as our step-by-step solution has led us to a contradiction (5 = 23). This suggests that there might be a misunderstanding of the equation's intended structure. The initial notation (18x = 30) : 6 = 90 - 67 is not standard and can be interpreted in different ways. We've explored one interpretation, where the division by 6 applies to the expression derived from solving 18x = 30, but this led to an inconsistency.
Another possible interpretation is that the entire left side, including the subtraction, is meant to be evaluated sequentially. In this case, we would first solve 18x = 30 for x, as we did before, which gives us x = 5/3. Then, we would substitute this value back into the expression and evaluate it according to the order of operations. However, even with this approach, the equation doesn't hold true. The discrepancy highlights the importance of clear notation in mathematical expressions.
When dealing with equations, it's essential to use parentheses and other symbols correctly to avoid ambiguity. If the equation was intended to be something different, such as 18x = (30 : 6) * (90 - 67), the solution process would be entirely different. In such a case, we would first simplify the right side: (30 : 6) * (90 - 67) = 5 * 23 = 115. Then, the equation would become 18x = 115, and we would solve for x by dividing both sides by 18: x = 115/18. This demonstrates how the placement of parentheses and the order of operations significantly impact the solution.
In conclusion, while we've made a diligent effort to solve the equation using the distributive property and step-by-step methods, the ambiguity in the original notation prevents us from arriving at a consistent solution. It's a valuable lesson in the importance of clear communication in mathematics. If you encounter similar equations, always clarify the intended order of operations to avoid confusion and ensure accurate solutions. Moving forward, always double-check the equation and its notation to ensure clarity and accuracy.
Key Takeaways and Best Practices
Let's wrap things up by summarizing the key takeaways and best practices we've learned from this exercise. First and foremost, the importance of clear and unambiguous notation in mathematical equations cannot be overstated. The initial equation (18x = 30) : 6 = 90 - 67 highlighted how a lack of clarity can lead to multiple interpretations and potential errors. Always use parentheses and other symbols to clearly indicate the order of operations and avoid any ambiguity. This is crucial not only for your understanding but also for anyone else who might be reading or working with your equations.
Secondly, understanding and applying the distributive property is a fundamental skill in algebra. While the direct application of the distributive property wasn't the primary focus in solving this particular equation due to its ambiguous nature, we did explore how the concept of distributing an operation (like division) can be approached. Remember, the distributive property is most commonly used to multiply a factor across a sum or difference within parentheses, such as a(b + c) = ab + ac. Practice applying this property in various contexts to strengthen your understanding.
Thirdly, a step-by-step approach is always beneficial when solving complex equations. Breaking down the problem into smaller, manageable steps allows you to address each operation systematically and reduce the likelihood of errors. In our solution, we first simplified both sides of the equation, then solved for x, and finally checked our solution. This methodical approach is a valuable habit to develop.
Lastly, don't hesitate to double-check your work and consider different interpretations if you encounter inconsistencies. In our case, the contradiction 5 = 23 indicated that there was likely an issue with the initial equation or our interpretation of it. This prompted us to re-evaluate our steps and consider alternative approaches. Being able to recognize and address potential errors is a crucial skill in problem-solving.
In summary, always strive for clarity in notation, master fundamental properties like the distributive property, adopt a step-by-step approach, and be prepared to re-evaluate your work if needed. By following these best practices, you'll be well-equipped to tackle a wide range of algebraic problems with confidence. Keep practicing, and you'll become a math whiz in no time!
