Unveiling The Subtraction Property Of Equality: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fundamental concept in algebra: the subtraction property of equality. We'll break down how it works and, more importantly, pinpoint the step in a given equation where it's applied. Let's get started!

Understanding the Subtraction Property of Equality

Alright, guys, let's talk basics. The subtraction property of equality is a rule that says if you subtract the same value from both sides of an equation, the equation remains balanced. Think of it like a seesaw: if you remove the same weight from both sides, the seesaw stays level. This property is crucial for solving equations because it allows us to isolate the variable, which is our ultimate goal. It helps us simplify the equation and get closer to finding the value of 'x'.

To put it simply, if a = b, then a - c = b - c. Where 'a' and 'b' represent expressions or values, and 'c' is the value we're subtracting from both sides. This ensures that the equality holds true. This is one of the most basic rules, and once you grasp the concept, you'll see how often it comes into play. It is very useful when we want to eliminate a number that is being added to a variable, like in the equation of this example.

This property is not just some abstract mathematical rule; it's a practical tool that allows us to solve a wide range of equations. Without it, or similar properties like addition, multiplication, and division properties of equality, solving equations would be incredibly difficult, if not impossible. So, understanding the subtraction property of equality is a key step in mastering algebra and problem-solving. This knowledge will set a strong foundation for more complex mathematical concepts.

Decoding the Equation: $\frac{1}{4}(12x + 8) + 4 = 3$

Now, let's apply the subtraction property of equality to solve a given equation. Our goal is to find the value of x. The equation we're working with is: $\frac{1}{4}(12x + 8) + 4 = 3$. This equation might look a little intimidating at first glance, but let's break it down step by step. We'll examine each step and identify where the subtraction property of equality is used. This process is not only about finding the correct answer but also about understanding the logic and reasoning behind each step.

Remember, in solving any equation, the aim is to isolate the variable. We can simplify our lives by following the order of operations in reverse. That is, we can remove the numbers that are added or subtracted first, before we remove the numbers that are multiplied or divided. This approach makes the process more organized and less prone to errors. So, let’s start breaking down the equation: $\frac{1}{4}(12x + 8) + 4 = 3$. Our initial step involves simplifying the left side of the equation. We’ll do this by distributing the $\frac{1}{4}$ across the terms inside the parentheses. After that, we must isolate the variable. Stay tuned and let’s unveil which step involves the subtraction property of equality!

Step-by-Step Solution Analysis

Here’s the step-by-step solution to the equation:

Now, let's analyze each step to identify where the subtraction property of equality is applied. Remember, the subtraction property involves subtracting the same value from both sides of the equation to maintain balance.

• Step 1: $3x + 2 + 4 = 3$

  In this step, we've simplified the equation by distributing $\frac{1}{4}$ across the terms in the parentheses: $\frac{1}{4} * 12x = 3x$ and $\frac{1}{4} * 8 = 2$. So, this step uses the distributive property and not the subtraction property of equality. This stage shows the transformation of the original equation into a more manageable form. We are essentially simplifying the left side of the equation, getting it ready for further simplification. So, we're not subtracting anything from both sides here; we're just applying the distributive property. It's like re-writing a sentence using different words to make it easier to understand without altering its meaning.

• Step 2: $3x + 6 = 3$

  Here, we've combined the constants on the left side: $2 + 4 = 6$. Again, no subtraction property is used here. We're simply simplifying by combining like terms. Combining like terms is a key simplification technique in algebra. Just imagine that we are organizing similar objects into a single group, so it’s easier to count them. It makes the equation simpler and easier to solve. This process makes the equation look cleaner, and it brings us closer to isolating the variable.

• Step 3: $3x = -3$

  Now, this is where the magic happens, guys! To isolate the term with x ($3x$), we need to remove the $+6$ from the left side of the equation. We do this by subtracting 6 from both sides of the equation. This is precisely the application of the subtraction property of equality! We subtract 6 from both sides to keep the equation balanced. If we do this, we get: $3x + 6 - 6 = 3 - 6$, which simplifies to $3x = -3$. This step is a direct application of the subtraction property. It demonstrates how subtracting the same value from both sides of an equation maintains the equality, bringing us closer to finding the value of x. It is a crucial step for achieving the equation’s final result.

• Step 4: $x = -1$

  Finally, in this step, we solve for x. We do this by dividing both sides of the equation by 3. This is using the division property of equality, not the subtraction property. Therefore, we are not interested in this step. This step provides the final answer to the question. It reveals the value of x that satisfies the initial equation. It is the culmination of all the previous steps, where the variable is successfully isolated.

The Answer: Which Step Shows the Subtraction Property?

So, the subtraction property of equality is applied in Step 3: $3x = -3$. This is where we subtract 6 from both sides of the equation to isolate the term with the variable. Knowing how to spot the application of these basic algebraic properties is key to quickly and accurately solving equations. You will see these properties being used very often when you solve algebra equations. Understanding each step in solving an equation helps you not only find the answer but also understand why the answer is correct.

Why is This Important?

Understanding the subtraction property of equality, along with other properties like the addition, multiplication, and division properties of equality, is fundamental for success in algebra. These properties are the building blocks that allow you to solve a wide variety of equations and tackle more complex mathematical problems. Mastering these concepts provides a solid foundation for higher-level mathematics. If you’re a student, being able to identify and apply these properties will help you excel in your math classes. If you are preparing for exams or standardized tests, knowing these properties will make your problem-solving process faster and more efficient.

Conclusion: Keep Practicing!

Alright, guys, that's it for today! We’ve successfully identified the step where the subtraction property of equality is used. Remember that practice is key, so keep working on those equations and strengthening your algebra skills! The more you practice, the more comfortable and confident you'll become in solving equations. Don't be afraid to try different problems and apply what you've learned. The journey of mastering algebra is a rewarding one, and each step brings you closer to greater mathematical understanding. So, keep up the fantastic work, and happy solving! If you have any further questions or want to delve deeper into these topics, don’t hesitate to ask! Stay curious, and keep exploring the amazing world of mathematics!