Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into solving the equation $2(9x - 8) = 8(x - 7)$. This is a classic algebra problem, and we'll break it down step by step to make sure everyone understands the process. Whether you're a seasoned pro or just starting out, this guide will help you confidently solve equations. We'll start by looking at the original equation and then systematically simplify and isolate x to find our solution. Let's get started!

Understanding the Basics of Equation Solving

Before we jump into the equation, let's refresh our understanding of what it means to solve an equation. An equation is simply a statement that two expressions are equal. Our goal is to find the value(s) of the variable (in this case, x) that make the equation true. In other words, we want to find the number that, when substituted for x, makes both sides of the equation equal. The core principle we'll use is to maintain the equality. Whatever operation we perform on one side of the equation, we must perform the exact same operation on the other side. This ensures that the equation remains balanced throughout the solution process. We'll use this principle of balance to manipulate the equation, gradually isolating x on one side and revealing its value. This is the cornerstone of solving equations, and it's essential to grasp this concept before we proceed. This foundation will make it easy to follow the steps and understand why we perform each operation. Remember, the ultimate aim is to find the value of x that satisfies the equation, and we achieve this by maintaining the equation's balance at every stage.

Why Solving Equations Matters

You might be wondering why solving equations is important. Well, equations are the language of mathematics and are used to model real-world scenarios. From calculating distances and speeds to understanding financial models and predicting scientific outcomes, equations are fundamental. They are also essential in fields like physics, engineering, computer science, and economics. Even in everyday life, you use equations, although you might not realize it. For example, when you budget, plan a trip, or cook using a recipe, you are implicitly using mathematical concepts to solve for an unknown (like the amount of ingredients you need). The ability to solve equations develops critical thinking and problem-solving skills, equipping you with valuable tools that extend far beyond the classroom. The more familiar you become with equations, the more capable you'll be in various aspects of life, making you a more informed and versatile individual. So, embrace the challenge, and let’s unlock the power of equations together!

Step-by-Step Solution

Now, let's solve the equation $2(9x - 8) = 8(x - 7)$ step by step.

Step 1: Distribute

The first step is to distribute the numbers outside the parentheses to the terms inside the parentheses. This means multiplying each term inside the parentheses by the number outside. Let's do this for both sides of the equation:

• On the left side: $2 * 9x = 18x$ and $2 * -8 = -16$. So, the left side becomes $18x - 16$.
• On the right side: $8 * x = 8x$ and $8 * -7 = -56$. So, the right side becomes $8x - 56$.

Our equation now looks like this: $18x - 16 = 8x - 56$. This distribution step simplifies the equation, allowing us to isolate the variable x more easily. It's the groundwork for the subsequent steps.

Step 2: Combine Like Terms

Next, we want to get all the x terms on one side of the equation and the constants (numbers without variables) on the other side. Let's start by subtracting $8x$ from both sides of the equation to move the $x$ term from the right side to the left side: $18x - 16 - 8x = 8x - 56 - 8x$. This simplifies to $10x - 16 = -56$.

Now, add 16 to both sides of the equation to move the constant term from the left side to the right side: $10x - 16 + 16 = -56 + 16$. This simplifies to $10x = -40$. Combining like terms is all about organizing the equation in a way that isolates the variable.

Step 3: Isolate the Variable

Now, we have a simplified equation: $10x = -40$. To isolate x, we need to get rid of the coefficient (the number multiplying x), which is 10. We do this by dividing both sides of the equation by 10: $(10x) / 10 = -40 / 10$. This gives us $x = -4$. This step is where we directly solve for x.

Step 4: Verify the Solution

Always a good idea to check your solution. Substitute x = -4 back into the original equation to ensure it holds true:

$2(9(-4) - 8) = 8((-4) - 7)$

$2(-36 - 8) = 8(-11)$

$2(-44) = -88$

$-88 = -88$

The solution checks out! This step verifies the accuracy of the solution. It is good practice.

Conclusion: The Answer

So, the correct answer is c.) $x = -4$. We've systematically solved the equation $2(9x - 8) = 8(x - 7)$ using distribution, combining like terms, and isolating the variable. This approach can be applied to many linear equations. Keep practicing, and you'll become a pro at solving equations in no time! Remember to always check your answers to verify the solution. The more you practice solving equations, the more comfortable and confident you will become. Good luck, and keep up the great work!

Quick Recap

• Original Equation: $2(9x - 8) = 8(x - 7)$
• Step 1 (Distribute): $18x - 16 = 8x - 56$
• Step 2 (Combine Like Terms): $10x = -40$
• Step 3 (Isolate x): $x = -4$
• Solution: $x = -4$

Hopefully, this detailed walkthrough helps you understand how to solve linear equations. Keep practicing, and you'll become a whiz in no time! Remember, the key is to stay organized and perform the same operations on both sides to keep the equation balanced.