Solving Equations Step-by-Step A Comprehensive Guide

Hey guys! Let's dive into solving equations, a fundamental skill in mathematics. We'll break down a specific example step-by-step, making sure we understand each move. This guide aims to not only show you how to solve equations but also why each step is necessary. So, grab your pencils and let’s get started!

Breaking Down the Equation

Let's take a look at the equation we're going to solve:

$\frac{t+2}{9} = 9$

Our goal here is crystal clear: we need to isolate 't' on one side of the equation. This means we want to get 't' all by itself so we can see what value makes the equation true. To do this, we'll carefully undo the operations that are affecting 't', step by step. Think of it like peeling an onion – we're removing layers until we get to the core.

Step 1: Clearing the Fraction

First things first, we need to tackle that fraction. Fractions can sometimes make equations look scary, but they're really not that bad! In our equation, we have 't+2' divided by 9. To get rid of this division, we need to do the opposite operation – multiplication. Remember, whatever we do to one side of the equation, we must do to the other side to keep things balanced. It's like a seesaw; if you add weight to one side, you need to add the same weight to the other to keep it level.

So, we multiply both sides of the equation by 9. This looks like:

9 * ($\frac{t+2}{9}$) = 9 * 9

On the left side, the multiplication by 9 cancels out the division by 9. This is exactly what we wanted! We're left with 't+2'. On the right side, 9 multiplied by 9 is 81. Our equation now looks much simpler:

t + 2 = 81

Why does this work? Multiplying by 9 is the inverse operation of dividing by 9. Inverse operations "undo" each other. Just like adding 5 and then subtracting 5 gets you back where you started, multiplying by 9 and then dividing by 9 cancels out. This is a crucial concept in solving equations.

Step 2: Isolating 't'

Now we're getting closer! We have 't + 2 = 81'. To get 't' completely alone, we need to get rid of the '+ 2'. What's the opposite of adding 2? Subtracting 2, of course! So, we subtract 2 from both sides of the equation:

t + 2 - 2 = 81 - 2

On the left side, the '+ 2' and '- 2' cancel each other out, leaving us with just 't'. On the right side, 81 minus 2 is 79. Our equation now looks like this:

t = 79

And there we have it! We've successfully isolated 't' and found its value. This means that if we substitute 79 for 't' in the original equation, the equation will be true. Let's double-check to be sure.

Putting It All Together: The Complete Solution

Let's recap the entire process to make sure we've got it all down:

1. Original Equation: $\frac{t+2}{9} = 9$
2. Multiply both sides by 9: This step gets rid of the fraction.
   • 9 * ($\frac{t+2}{9}$) = 9 * 9
   • t + 2 = 81
3. Subtract 2 from both sides: This isolates 't'.
   • t + 2 - 2 = 81 - 2
   • t = 79

So, the solution to the equation is t = 79.

Key Descriptions Missing (and Why They're Important)

In the original problem setup, we were asked to fill in missing descriptions for the steps. Here are the descriptions that should be included, along with why they are essential for understanding the solution:

• A. Multiply both sides by 9: This description clearly states the operation performed in the first step. It's crucial because it explains how we eliminated the fraction, which is a common first step in solving equations involving fractions. Understanding why we multiply by 9 (to undo the division) is just as important as knowing that we multiply by 9.
• B. Subtract 2 from both sides: This description explains the action taken to isolate 't'. It highlights the inverse operation concept – subtracting to undo addition. This step is the final move in getting 't' by itself and finding its value.

By explicitly stating these descriptions, we solidify our understanding of the equation-solving process. It's not just about the steps; it's about the reasoning behind them.

Why This Matters: The Bigger Picture

Solving equations is not just a math class exercise; it's a fundamental skill that's used in countless real-world situations. From calculating finances to designing structures to understanding scientific phenomena, the ability to manipulate equations is essential.

Think about these scenarios:

• Budgeting: You might need to solve an equation to figure out how much you can spend each month while still saving a certain amount.
• Cooking: If you want to double a recipe, you'll need to solve equations to adjust the ingredient amounts.
• Travel: Calculating travel time or distance often involves solving equations.
• Science and Engineering: Equations are the language of science and engineering. They're used to model everything from the motion of planets to the flow of electricity.

The skills you learn in solving simple equations like the one we tackled here are the building blocks for more complex mathematical concepts. As you progress in your studies, you'll encounter more challenging equations, but the underlying principles remain the same. Understanding how to isolate variables and use inverse operations will serve you well in algebra, calculus, and beyond.

Tips and Tricks for Equation-Solving Success

Here are a few extra tips to help you become an equation-solving pro:

1. Always show your work: Writing down each step helps you keep track of your progress and makes it easier to spot mistakes. It also makes your solution clear for anyone else to follow.
2. Check your answer: Once you've found a solution, plug it back into the original equation to make sure it works. This is a great way to catch errors and build confidence in your answer.
3. Practice, practice, practice: The more you practice solving equations, the better you'll become. Start with simple equations and gradually work your way up to more challenging ones.
4. Don't be afraid to ask for help: If you're stuck on a problem, don't hesitate to ask your teacher, a tutor, or a classmate for assistance. Math can be tough, and everyone needs help sometimes.
5. Look for patterns: As you solve more equations, you'll start to notice patterns and shortcuts. This will make you a faster and more efficient problem-solver.
6. Understand the “Why”: It’s not enough to just memorize steps. Try to understand why each step works. This deeper understanding will make you a more flexible and confident equation solver.

Let's Summarize: Key Takeaways

• Solving equations involves isolating the variable. We want to get the variable all by itself on one side of the equation.
• Use inverse operations. To undo an operation, use its inverse. Addition and subtraction are inverse operations, as are multiplication and division.
• Keep the equation balanced. Whatever you do to one side of the equation, you must do to the other side.
• Show your work and check your answers. These are crucial habits for success.

Conclusion: You've Got This!

Solving equations might seem daunting at first, but with practice and a clear understanding of the steps involved, you can master this important skill. Remember to break down the problem into smaller parts, use inverse operations, and always keep the equation balanced. And most importantly, don't give up! With persistence, you'll be solving equations like a pro in no time. Keep up the great work, guys!