Solving Equations: Justify Each Step

Hey Plastik Magazine readers! Today, we're diving into the world of equations and how to solve them step-by-step. But we’re not just solving; we’re going to justify each move we make. Think of it like explaining your strategy in a game – every action needs a reason. Let's break down the equation (1/4)x - (9/4)x - 7 = -15 and see why each step is crucial.

Step 1: The Given Equation

Our starting point is the equation itself: (1/4)x - (9/4)x - 7 = -15. This is what we call the "given" because, well, it's given to us! There's no magic or mystery here; it’s simply the problem we're tasked to solve. Before we even begin manipulating the equation, we need to understand what it represents. This equation is a linear equation, meaning it involves a variable (x) raised to the power of 1. Our goal is to isolate x on one side of the equation to find its value. This initial step of acknowledging the given equation sets the stage for all the subsequent steps. It's like the foundation of a building; without it, the rest of the structure wouldn't stand. Understanding the given also allows us to identify the different terms and operations involved, which is essential for deciding the best approach to solve the equation. This approach includes combining like terms, isolating the variable, and using inverse operations to maintain the equality. It is important to double-check the given equation to ensure accuracy. Any errors in the initial equation will propagate through the rest of the steps, leading to an incorrect solution. So, always make sure you have the correct starting point before proceeding.

Step 2: Combining Like Terms

Now, let's simplify things. We've got (1/4)x and (-9/4)x. These are like terms because they both contain the variable 'x'. Combining them is like adding apples to apples – or, in this case, fractions of x to fractions of x. So, (1/4)x - (9/4)x becomes (-8/4)x, which simplifies to -2x. Our equation now looks like this: -2x - 7 = -15. The justification here is the distributive property in reverse and combining like terms. Essentially, we're using the fact that ac + bc = (a+b)*c. In our case, c is 'x', a is 1/4, and b is -9/4. Combining like terms helps to simplify the equation and make it easier to solve. By reducing the number of terms, we reduce the number of operations we need to perform. This not only makes the process more efficient but also less prone to errors. When combining like terms, it's crucial to pay attention to the signs. Make sure you're adding or subtracting the coefficients correctly. A simple mistake in this step can lead to a wrong answer. Also, remember that you can only combine terms that have the same variable and exponent. For example, you can't combine -2x with -7 because -7 is a constant term without any variable. This simplification is a fundamental algebraic technique used to solve many types of equations.

Step 3: Isolating the Variable Term

We want to get the '-2x' term all by itself on one side of the equation. To do this, we need to get rid of the '-7'. How? By adding 7 to both sides of the equation. This is based on the addition property of equality, which states that if you add the same value to both sides of an equation, the equation remains balanced. So, -2x - 7 + 7 = -15 + 7, which simplifies to -2x = -8. This step is crucial because it brings us closer to isolating the variable 'x', which is our ultimate goal. The addition property of equality is a fundamental concept in algebra, ensuring that we maintain the balance of the equation while manipulating it. Remember, whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This keeps the equation true and allows us to isolate the variable without changing its value. Adding 7 to both sides cancels out the -7 on the left side, leaving us with just the term containing 'x'. This is a key step in solving for 'x' because it separates the variable term from the constant terms. By isolating the variable term, we can then proceed to divide both sides by the coefficient of 'x' to find its value.

Step 4: Solving for x

Now we have -2x = -8. To find the value of x, we need to get rid of the '-2' that's multiplying it. We do this by dividing both sides of the equation by -2. This is based on the division property of equality, which, like the addition property, states that if you divide both sides of an equation by the same non-zero value, the equation remains balanced. So, (-2x) / -2 = (-8) / -2, which simplifies to x = 4. The justification here is the division property of equality. This is the final step in solving for x, and it reveals the value that makes the original equation true. The division property of equality is another fundamental concept in algebra, ensuring that we maintain the balance of the equation while isolating the variable. Dividing both sides by -2 cancels out the -2 on the left side, leaving us with 'x' by itself. This gives us the value of 'x', which is 4. This means that if we substitute 4 for 'x' in the original equation, the equation will hold true. Always double-check your solution by plugging it back into the original equation to make sure it satisfies the equation. This is a good way to catch any mistakes you might have made along the way. In this case, plugging 4 into the original equation gives us (1/4)(4) - (9/4)(4) - 7 = 1 - 9 - 7 = -15, which confirms that our solution is correct.

Summary of Justifications

Let's recap the reasons for each step:

• Step 1: Given (the original equation)
• Step 2: Combining Like Terms (using the distributive property in reverse)
• Step 3: Addition Property of Equality (adding 7 to both sides)
• Step 4: Division Property of Equality (dividing both sides by -2)

And there you have it! By understanding the reasons behind each step, solving equations becomes less about blindly following rules and more about logical, justified moves. Keep practicing, and you'll be an equation-solving pro in no time! Remember to always double-check your work and stay curious. Keep exploring the world of mathematics, and you'll find that it's full of fascinating patterns and relationships.