Solving Equations: Justify Each Step Correctly

Hey Plastik Magazine readers! Let's dive into the world of algebra and make sure we can solve equations like pros. This is going to be super helpful, trust me! Today, we're not just going to solve, but we're going to understand why each step works. This is like, super important for building a solid foundation in math. We'll break down how to justify each step when solving an equation, making sure everything is clear as day. Get ready to flex those math muscles, guys!

The Equation's Journey: A Step-by-Step Guide

Alright, let's get down to business. We are going to analyze this equation step by step, which will help us understand the process. We will show you how to justify each step along the way. Think of it like a detective story, where each step reveals a clue. We start with the given equation: $13x - 4 = -9(4 + 3x) + 12$. Our goal is to isolate x and find its value. So, here's how we break it down, making sure to justify each step.

Step 1: The Foundation - Given

Our journey begins with the given equation: $13x - 4 = -9(4 + 3x) + 12$. This is our starting point, our initial condition. It's the information we're provided, and we haven't done anything to it yet. Think of it like the first scene in a movie – the setup. This step requires no justification because it is simply stating the equation we will be working with. We are not doing anything to change the equation. We’re just writing down what we start with. It is crucial to have the original equation so that you can follow along with each of the transformations. So, there is no need to overthink this step. Just copy the equation as it is, and then you'll move to the next. Easy, right? It sets the stage for everything that follows. Make sure you don't miss this crucial beginning. It's like the very first domino in a long chain reaction.

Step 2: Distributing the Love – Distributive Property

Next, we tackle the right side of the equation. We see $-9(4 + 3x)$. This is where the Distributive Property comes in handy. Remember, the Distributive Property states that a(b + c) = ab + ac. So, we multiply -9 by both 4 and 3x: $-9 * 4 = -36$ and $-9 * 3x = -27x$. Our equation now looks like this: $13x - 4 = -36 - 27x + 12$. This step simplifies the expression by removing the parentheses. By applying this property, we’re essentially expanding the expression, making it easier to work with. It's like opening up a box to see what's inside. The Distributive Property is a fundamental concept in algebra, and understanding it is key to solving equations. It allows us to simplify and manipulate expressions effectively. This step is a necessary process to remove the parentheses, and to work with the terms inside.

Step 3: Combining Like Terms – Simplifying

Now, let's tidy things up a bit. We have $-36$ and $+12$ on the right side of the equation, which are constants (numbers without variables). We can combine these like terms to simplify the equation. $-36 + 12 = -24$. This gives us $13x - 4 = -24 - 27x$. Combining like terms is a crucial part of simplifying expressions. It’s like sorting your clothes: you put all the shirts together, all the pants together, and so on. This makes it easier to manage and understand what you have. This simplification keeps our equation clean and makes it easier to work with. This is a crucial step to solve for x. You want to get the constants all on one side of the equation.

More Steps to Solve for X

Now we're cruising, and we will continue our goal to justify each step, this time with a more focused approach. Remember, our aim is to isolate x. We will perform mathematical operations to change the equation, to get x alone.

Step 4: Moving the x Term – Addition Property of Equality

We want all the x terms on one side of the equation. Let’s move the $-27x$ from the right side to the left side. To do this, we add $27x$ to both sides. Why? Because the Addition Property of Equality says that if we add the same value to both sides of an equation, the equation remains balanced. So, we add $27x$ to both sides. Our equation becomes $13x + 27x - 4 = -24$. Simplifying, we get $40x - 4 = -24$. This step is a cornerstone in solving equations. The Addition Property of Equality is a fundamental rule that ensures we maintain the balance of the equation. Without it, our answer would be incorrect. This step brings all our x terms together and makes our equation more manageable. It is an extremely important step that ensures a balanced equation.

Step 5: Isolating the Constant – Addition Property of Equality

Next, we need to get rid of the $-4$ on the left side. To do this, we add $4$ to both sides, again using the Addition Property of Equality. This cancels out the $-4$ on the left, leaving us with $40x = -20$. We are getting closer to isolating x. By adding 4 to both sides, we isolate the x term, and then move towards getting x by itself. Each of these steps moves us closer to a solution. This step is about isolating the constant terms on one side of the equation, making it easier to solve for x. You need to keep doing the same operation on both sides to keep the equation balanced.

Step 6: The Final Step – Division Property of Equality

Almost there! We have $40x = -20$. To isolate x, we need to get rid of the 40 that's multiplying it. We do this by dividing both sides by 40. This is the Division Property of Equality, which states that if we divide both sides of an equation by the same non-zero number, the equation remains balanced. So, $-20 / 40 = -1/2$. Thus, $x = -1/2$. This step is the grand finale. By dividing both sides by 40, we isolate x and find its value. The Division Property of Equality is just as crucial as the Addition Property. Both are essential for maintaining the integrity of our equation. Make sure you don't forget to do the same operation to both sides of the equation. You are left with the solution to the equation.

Conclusion: You Got This!

Alright, we've walked through solving the equation step-by-step, making sure to justify each step along the way. From the initial setup with the equation to the final solution, we've used several fundamental properties. Remember, understanding why each step works is just as important as getting the right answer. The distributive property, the properties of equality, combining like terms – these are your tools. Math can be tricky, but by taking things step-by-step and understanding the underlying concepts, you can solve any equation. Keep practicing, and you'll be acing these problems in no time! So, what do you think, guys? Ready to try some on your own? Let me know if you need more tips or examples. Keep up the great work!