Solving Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into a classic math problem: $x + (3x + 20) = 180$. Don't worry, it's not as scary as it looks! We'll break down each step, explaining the properties used along the way. Think of it like a fun puzzle – we'll figure out what "X" is together! Get ready to flex those math muscles and understand the why behind each move. This step-by-step guide is designed to make you feel like a math whiz in no time. We will transform complex equations into something you can easily follow, breaking down each step to make it as simple as possible. By the end, you'll be able to solve these types of equations and confidently explain your reasoning. You'll gain a solid understanding of how mathematical properties work. So, grab your pencils and let's get started. We are going to reveal the fascinating world of equations and the principles that govern them. This is more than just math; it is about developing problem-solving skills that can be applied to many aspects of your life. This guide will help you understand the core concepts. Ready to unlock the secrets behind equations? Let's go!

Step 1: Combining Like Terms

Our first step in solving $x + (3x + 20) = 180$ involves simplifying the left side of the equation. We need to combine the like terms. Remember, like terms are terms that have the same variable raised to the same power. In our equation, we have "x" and "3x". To combine them, we simply add their coefficients (the numbers in front of the variables). "x" is the same as "1x", so we are adding 1x and 3x. Adding these together, we get 4x. The constant term "20" stays as it is since there are no other constants to combine it with. This step is about making the equation easier to handle by reducing the number of terms. The simplified equation after combining like terms will then be $4x + 20 = 180$. This simplification is critical because it reduces the complexity of the equation, making the subsequent steps easier to execute and understand. By carefully combining the terms, we streamline the process and pave the way for solving for "x" more efficiently. The main idea here is to make the equation simpler and less cluttered, allowing us to focus on isolating "x". This step is all about tidying up the equation so we can proceed with confidence. This prepares us for the next phase, where we will solve for the unknown variable.

Property Used: Simplify

The property we use in this step is Simplify. Simplify means to reduce the equation to its simplest form. It is the core of making the equation manageable by combining similar components and getting rid of unnecessary things, as shown above. This involves grouping similar terms together, making the equation more concise and easier to solve. When we "simplify," we apply basic algebraic rules to make sure our equations become as easy as possible to deal with. This property helps streamline the problem so that it is less complex, which makes the whole process easier.

Step 2: Isolating the Variable

Now that we've simplified the equation to $4x + 20 = 180$, the next step is to isolate the term with the variable "x". Our goal is to get the "x" term by itself on one side of the equation. To do this, we need to remove the constant term "20" that is added to the "4x". We use the Subtraction Property of Equality for this. This property states that if we subtract the same number from both sides of the equation, the equation remains balanced. We subtract "20" from both sides. On the left side, "20" cancels out, leaving us with just "4x". On the right side, "180 - 20" gives us 160. Following this step we'll end up with $4x = 160$. Getting to this stage means we are very close to finding the value of "x". It involves removing any numbers that are added or subtracted from the variable term. This isolation is crucial to finding the exact value of the variable. By focusing on getting the variable term alone, we make it simple to find what "x" is. This step is all about stripping away everything except the variable to prepare it for the final stage. The careful use of the Subtraction Property guarantees that the equation stays balanced. This balancing act ensures that any operation we do on one side is done on the other to preserve the equation's truth.

Property Used: Subtraction Property of Equality

The Subtraction Property of Equality says that if we subtract the same number from both sides of an equation, the equality remains true. In our case, we subtracted 20 from both sides. It is fundamental in algebra and ensures that any action performed on one side of the equation is mirrored on the other side. This is like maintaining balance on a scale; to keep the scale balanced, you must add or remove the same weight on both sides. This property is vital for ensuring the equation stays valid as we simplify it. It is what allows us to "move" terms from one side of the equation to the other without changing the solution. This is a cornerstone for solving many types of equations, and it is crucial for our work. The Subtraction Property of Equality is a straightforward idea with major implications. By following this property, we guarantee the integrity of our calculations and can solve for "x" correctly.

Step 3: Solving for x

We've successfully simplified and isolated the variable term. Now we're at the final step: solving for "x". We have the equation $4x = 160$. Here, "4" is multiplied by "x". To get "x" by itself, we need to do the opposite operation: division. We use the Division Property of Equality. This property states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. We divide both sides of the equation by "4". On the left side, "4x / 4" simplifies to "x". On the right side, "160 / 4" equals 40. This leaves us with the answer: $x = 40$. We've solved the equation! Getting "x" by itself means we've found the solution. It involved undoing the multiplication and getting to the heart of the matter. This step is the culmination of all our work. By following the Division Property, we ensure that the value of "x" is correct. This is the moment when all the calculations come together, giving us a clear result. At this stage, it all falls into place. The end result is what we've been working toward. This provides the final answer to the equation.

Property Used: Division Property of Equality

The Division Property of Equality states that if you divide both sides of an equation by the same non-zero number, the equation remains true. This is the counterpart to the Multiplication Property of Equality. This property is like the reverse of multiplication. It makes sure that equations are always balanced as we work on them. It is another way to modify equations without affecting their answers. By dividing both sides by the same amount, we make sure that both sides stay in proportion. We use this principle to isolate the variable so that we can find its value. By keeping both sides in balance, we are guaranteed an accurate outcome. The Division Property of Equality plays a key role in finding the answers to equations. The use of this property shows how to accurately solve equations and find their answers.

Summary of Steps and Properties

Let's recap what we've done and the properties we've used:

1. Original Equation: $x + (3x + 20) = 180$
2. Combine Like Terms: $4x + 20 = 180$ (Property: Simplify)
3. Subtract 20 from both sides: $4x = 160$ (Property: Subtraction Property of Equality)
4. Divide both sides by 4: $x = 40$ (Property: Division Property of Equality)

So, the correct answers are:

A. Substitution Property (Not used in this solution) B. Angle Addition Postulate (Not applicable to this algebraic equation) C. Division Property of Equality (Used in Step 3) D. Simplify (Used in Step 1) E. Subtraction Property of Equality (Used in Step 2)

Great job, everyone! You've successfully solved an algebraic equation and learned about the properties used in the process. Keep practicing, and you'll become a math pro in no time! Remember, understanding the why behind each step is as important as getting the right answer. Keep up the excellent work, and enjoy your math journey! Keep those math skills sharp, and you'll be able to solve increasingly complex problems.