Solve For 'd': Mastering Linear Equations

Hey math enthusiasts! Today, we're diving into the world of solving for 'd' in a linear equation. Don't worry, it's not as scary as it sounds! We're going to break down the equation $1.8(6.4d + 4) = 1.6(8.3d + 6.7)$ step by step, making sure you grasp every concept along the way. Get ready to flex those algebra muscles! Let's get started!

Understanding the Basics: Linear Equations and 'd'

Before we jump into the equation, let's chat about what we're dealing with. A linear equation is like a straight line on a graph. It's an equation where the highest power of the variable (in this case, 'd') is 1. Our goal is to find the value of 'd' that makes the equation true. Think of it like a puzzle – we're trying to find the missing piece, which is the value of 'd' that balances both sides of the equation. In our equation, $1.8(6.4d + 4) = 1.6(8.3d + 6.7)$, we have 'd' hidden inside parentheses, and it's multiplied by numbers. The key to solving this is to isolate 'd' on one side of the equation.

So, what does it mean to solve for 'd'? Simply put, it means we want to find the numerical value that 'd' represents. This number will make the equation true when we substitute it back into the original expression. The process involves using algebraic operations – such as multiplication, division, addition, and subtraction – to manipulate the equation. The key is to do the same thing to both sides of the equation. This ensures that the balance is maintained, and we don't change the truth of the equation. We are essentially unwrapping the variable 'd' from all the numbers and operations surrounding it until it stands alone, ready to reveal its value. This entire process is about isolating the variable and uncovering its true identity within the equation. This is the heart of what we will be doing, and you'll become a pro at it by the end of this guide.

Now, let's move on to the actual equation. We will be using the concepts of distribution, combining like terms, and isolating the variable. Remember, the goal is always to get 'd' by itself on one side of the equal sign. So, are you ready to solve the puzzle? Get your pencils ready, because we're about to crack this equation!

Step-by-Step Solution: Unveiling the Value of 'd'

Alright, guys, let's get down to business and solve $1.8(6.4d + 4) = 1.6(8.3d + 6.7)$. We'll break it down into easy-to-follow steps.

Step 1: Distribute

The first step is to get rid of those pesky parentheses using the distributive property. This means multiplying the number outside the parentheses by each term inside the parentheses. So:

• $1.8 * 6.4d = 11.52d$
• $1.8 * 4 = 7.2$
• $1.6 * 8.3d = 13.28d$
• $1.6 * 6.7 = 10.72$

Now, our equation looks like this: $11.52d + 7.2 = 13.28d + 10.72$.

Step 2: Combine 'd' terms

Next, we want to get all the 'd' terms on one side of the equation. Let's subtract $11.52d$ from both sides:

$11.52d + 7.2 - 11.52d = 13.28d + 10.72 - 11.52d$

This simplifies to:

$7.2 = 1.76d + 10.72$

Step 3: Isolate 'd'

Now, let's get the constant terms (the numbers without 'd') on the other side. Subtract $10.72$ from both sides:

$7.2 - 10.72 = 1.76d + 10.72 - 10.72$

This gives us:

$-3.52 = 1.76d$

Step 4: Solve for 'd'

Finally, to solve for 'd', divide both sides by $1.76$:

$-3.52 / 1.76 = 1.76d / 1.76$

This simplifies to:

$d = -2$

And there you have it! We've found that $d = -2$. We have successfully navigated through the steps of distribution, combining like terms, and isolating the variable. By sticking to these processes and keeping things balanced, we have solved for 'd'. Wasn't that fun?

Verification: Checking Our Answer

Always, always, always check your answer! This is a crucial step to make sure you haven't made any mistakes along the way. Let's plug $d = -2$ back into the original equation:

$1.8(6.4(-2) + 4) = 1.6(8.3(-2) + 6.7)$

Simplify within the parentheses:

$1.8(-12.8 + 4) = 1.6(-16.6 + 6.7)$

$1.8(-8.8) = 1.6(-9.9)$

$-15.84 = -15.84$

It checks out! Our solution is correct. This is awesome because it indicates we understood all the steps in the equation, and we successfully navigated them. This makes it a great accomplishment, and now we know that $d = -2$ is indeed the correct answer. The left side equals the right side. This step will help you to verify all your answers, and it helps you to be a pro at solving equations.

Tips and Tricks: Mastering Linear Equations

Here are some handy tips and tricks to help you become a linear equation superstar:

• Stay Organized: Write out each step clearly. Don't try to do too much in your head. Write down every step, it helps!
• Double-Check Your Work: Mistakes happen! Always check your answer by plugging it back into the original equation.
• Practice Makes Perfect: The more you practice, the easier it becomes. Do lots of example problems. You can find tons of online resources.
• Know Your Properties: Understand the distributive property, the commutative property, and the associative property. These are your best friends in algebra.
• Simplify First: If possible, simplify each side of the equation before you start manipulating it.
• Watch Out for Signs: Be extra careful with positive and negative signs. A small mistake can lead to a wrong answer. Take your time, and be focused!

Common Mistakes and How to Avoid Them

Let's be real, even the best of us make mistakes. Here are some common pitfalls and how to avoid them:

• Incorrect Distribution: Make sure you multiply the number outside the parentheses by every term inside.
• Sign Errors: Double-check your signs, especially when subtracting negative numbers.
• Combining Unlike Terms: You can only combine terms that have the same variable and exponent (e.g., you can't add 'd' and a constant number directly). Remember to only combine like terms!
• Forgetting to Check Your Answer: This is a big one! Always verify your solution to catch any errors.

By being aware of these potential pitfalls, you can navigate your equation with confidence and accuracy. Remember, practice and attention to detail are your best allies.

Beyond the Basics: Expanding Your Algebra Skills

Once you're comfortable solving these types of equations, you can explore more advanced topics, such as systems of equations, inequalities, and quadratic equations. Remember, each skill builds upon the previous one. Keep at it, and you'll be amazed at what you can achieve. Also, don't be afraid to seek help when you need it. There are lots of online resources, your teachers, and even your friends to help you navigate through these equations.

• Systems of Equations: Learn how to solve for multiple variables using techniques like substitution and elimination.
• Inequalities: Understand how to solve and graph inequalities, which involve symbols like <, >, ≤, and ≥.
• Quadratic Equations: Dive into equations with variables raised to the power of 2. These can be solved using factoring, completing the square, or the quadratic formula.

Conclusion: You've Got This!

Congratulations! You've successfully navigated the equation and solved for 'd'. Remember, solving linear equations is a fundamental skill in mathematics, and with practice, you'll become a master. Keep practicing, stay organized, and don't be afraid to ask for help. You've got this!

I hope you found this guide helpful and easy to understand. Now, go forth and conquer those equations! Good luck with your math endeavors, and remember to have fun along the way! Happy solving, folks!