Solving Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into the world of solving equations, a fundamental skill in mathematics. We'll walk through a specific example, filling in the missing pieces and understanding the reasoning behind each step. Get ready to flex those equation-solving muscles! Remember, mastering this will give you a major advantage in tackling all sorts of math problems. The process might seem daunting at first, but trust me, with practice, it becomes second nature. We're going to break down the problem bit by bit so you can easily understand each step. We will also address how to simplify fractions, which is a crucial part of getting the right answer and making it look nice and tidy.

Let's get started. We're going to be focusing on the problem in the table provided, where we will fill in the missing terms and give the appropriate descriptions. The goal is not just to find the answer, but to understand why each step is taken. This approach is key to developing a strong foundation in math, helping you tackle more complex problems with confidence. So, let’s get into the specifics of solving equations! I will use a very straightforward, step-by-step method and explain it in detail. Keep your eyes on the ball, and you will become proficient in solving the problems very soon. With each completed step, you are one step closer to solving the equation. The more you practice, the easier it becomes. I want to tell you that there is no shortcut, so practice, and you will become good at it. You will find that the more you practice these types of problems, the easier it gets to see the relationships between numbers and variables.

We start with the equation and proceed systematically, ensuring that we maintain the equality throughout the process. This meticulous approach is important for maintaining accuracy. Remember, the goal is always to isolate the variable, which, in this case, is t. We want to get t by itself on one side of the equation. This is like unraveling a puzzle; with each step, we get closer to the solution. We will use the given format to make our explanations very clear. It’s like a treasure hunt; we have clues (the equations), and our aim is to find the treasure (the value of t). Each step we take is designed to bring us closer to the treasure, making sure we stay on track. We'll be using different arithmetic operations. This might seem a little complicated, but with time, you will learn these operations and procedures, and it will be as easy as reciting your ABCs. You can think of it as a dance, where each move (operation) changes the equation but keeps the balance intact. Think of the equals sign as the center point of the balance. Our goal is to maintain the balance throughout the process.

Step-by-Step Solution

Here’s the equation we'll be solving:

Let's break it down step by step:

Step 1: Combining Like Terms

Alright, in the first step, we simply combine the constant terms on the left side of the equation. Both 2 and 7 are just numbers, so we can add them together. This simplifies the equation and makes the next steps easier. Always start by simplifying as much as you can. This also applies to bigger equations, the more you simplify in each step, the easier it will be to solve the equation. The good thing about these problems is that you can always check your answer. Once you are done solving the equation, take the answer and put it back into the original equation to see if it makes sense. If both sides of the equation equal each other, then you know you did it correctly!

When we combine these two terms, we get 9. This means that we are going to replace the 2 and 7 with a 9 in the equation. Think of it as consolidating similar items – you combine the 2 apples with the 7 apples, and now you have 9 apples total. We are not doing anything too fancy; we are just making the equation a little bit cleaner. By combining these terms, we make the equation more manageable. We are just using the basic principles of addition. There is nothing tricky about it. So, we now have $5t + 9 = 9$. We are not moving the terms to the other side of the equation at this step, we are only combining them. Keeping the process easy to understand is the key here. The more you familiarize yourself with these kinds of equations, the easier it gets.

Step 2: Isolating the Variable

Next, we need to get the term with t by itself. To do this, we need to get rid of that pesky ‘+ 9’. To get rid of the +9, we need to do the opposite operation: subtraction. This is where the balance of the equation comes in. If we subtract 9 from the left side, we must subtract 9 from the right side as well. This guarantees that the equation remains balanced. It’s like having a scale; to keep it balanced, you have to add or remove the same amount from both sides. We perform the same operation on both sides to maintain equality. You always need to keep the sides balanced. Remember, what we do to one side of the equation, we must do to the other. Doing this, we're left with $5t = 0$.

This simple act of subtracting keeps the equation true. At this point, we will not simplify any fractions; our goal here is to isolate the variable. The variable t is getting closer to being alone. We are slowly but surely getting closer to the solution. Each step has its own purpose, and we are working towards the final solution. The key here is to keep the equation balanced. The most common mistake beginners make is when they perform an operation to one side and forget to do it to the other. Just remember that whatever operation you perform on one side, you must perform it on the other. That is the only way to keep the equation balanced. This step is a critical component to achieving the solution to the equation.

Step 3: Solving for the Variable

Almost there! Now we have $5t = 0$. In order to find what one t is equal to, we will divide both sides by 5. The key is to isolate t. We are dividing because the variable t is being multiplied by 5. To remove the coefficient of t (which is 5), we divide both sides by 5. Once again, we’re maintaining the balance. Just remember, in order to keep the balance, we need to perform the same operation on both sides of the equation. So, if we divide the left side by 5, we have to divide the right side by 5 as well. If we divided just one side of the equation, the equation would be incorrect. This operation will isolate the t. This gives us the final answer. In this case, we get t = 0.

This step gives us the value of the variable t. Congratulations, you did it! Now, the equation is solved, and we've found the solution. Always remember to check your work; it's a good habit to verify that the answer makes sense in the original equation. Let’s make sure our answer makes sense. When in doubt, always go back to the original equation and plug in the value of t that we found. By substituting 0 for t, we have: $5 * 0 + 2 + 7 = 9$. This simplifies to $0 + 2 + 7 = 9$, or $9=9$. Since the left side of the equation equals the right side, we know we have the correct answer. You can pat yourself on the back, as we've successfully navigated the process of solving equations! The process that we followed here can be applied to many other equations. Keep practicing, and you'll find that solving equations becomes easier and easier. Each time you solve an equation, it is going to get easier. It is like exercising; the more you exercise, the easier it gets. The key is to practice every day.

Conclusion

And that's the whole process, guys! We started with an equation, step by step, we combined terms, isolated the variable, and then solved for it. The beauty of this is that the same process can be applied to a variety of equations. Always remember to maintain the balance of the equation. Keep practicing, and you'll become a pro at solving equations in no time! Keep in mind that understanding each step is more important than simply finding the answer. When you understand the logic behind each step, you can apply this to more complex equations. Good luck, guys, and happy equation solving!