Solving Equations: A Step-by-Step Guide

Hey everyone! Let's dive into solving the equation $4(4c-3)=2(5c+18)$. This might seem a little daunting at first, but trust me, with a few simple steps, we'll crack it! This guide will break down the process in a friendly, easy-to-understand way, so whether you're a math whiz or just starting out, you'll be able to follow along. We will cover the basic steps required to solve equations effectively. We will simplify and find the solution for the provided formula. Let's get started, shall we?

Understanding the Basics: What is an Equation?

Before we jump into the equation, let's make sure we're all on the same page about what an equation actually is. An equation is simply a mathematical statement that shows that two expressions are equal. Think of it like a balanced scale: whatever you do to one side, you must do to the other to keep it balanced. In our case, the equation $4(4c-3)=2(5c+18)$ tells us that the expression on the left-hand side ($4(4c-3)$) has the same value as the expression on the right-hand side ($2(5c+18)$). Our goal is to find the value of the variable 'c' that makes this statement true. The variable, represented by 'c' in this equation, is a symbol for a number we don't know yet. To solve an equation, we must isolate the variable, which means getting it by itself on one side of the equation. This involves using various algebraic operations, like addition, subtraction, multiplication, and division, to manipulate the equation while maintaining its balance. The aim is to find the value of the variable that makes the equation true. Understanding the concept of an equation as a balanced scale is crucial for solving equations because it helps us remember that any operation we perform on one side of the equation must also be performed on the other side to maintain equality. Let's consider a simple example like x + 2 = 5. To isolate 'x', you would subtract 2 from both sides, resulting in x = 3. This approach is fundamental in dealing with more complex equations like the one we're solving today. Let's delve into the specific equation at hand. Understanding these fundamentals helps in handling more intricate equations like the one we're working with today.

The Importance of Order of Operations

One crucial element when working with equations is understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). PEMDAS provides a standardized approach to evaluating mathematical expressions, ensuring that you arrive at the correct answer. The order is as follows: first, perform calculations within Parentheses (or brackets); then, calculate any Exponents (powers and roots); next, perform Multiplication and Division from left to right; and finally, perform Addition and Subtraction from left to right. It is imperative to perform these operations in the prescribed sequence. A common mistake is to overlook the order of operations, which can lead to errors. For example, in our equation, we need to address the parentheses first by distributing the numbers outside them. The correct application of PEMDAS is essential for accuracy in solving mathematical problems. For instance, in the equation 2 * (3 + 4), you must first perform the addition within the parentheses (3 + 4 = 7) and then multiply by 2, resulting in 14. If you were to multiply 2 by 3 first, you'd end up with an incorrect answer. Keeping this order in mind is key to avoiding these common errors. Throughout this guide, we'll continually emphasize the application of the order of operations to ensure clarity. It is a fundamental principle that guides all our calculations and helps to maintain the equation's integrity.

Step-by-Step Solution: Unraveling the Equation

Alright, guys, let's get down to the nitty-gritty and solve $4(4c-3)=2(5c+18)$ step by step. I'll break down each action so you won't miss a beat. We're going to use the distributive property first, then simplify, and finally isolate the variable. This will allow us to find the correct answer easily. Each action will be explained clearly. Here is the step-by-step process of our equation.

Step 1: Distribute and Expand

First things first, we need to get rid of those parentheses. To do this, we'll use the distributive property. This means multiplying the number outside the parentheses by each term inside the parentheses. So, for the left side of the equation, we'll multiply 4 by both 4c and -3. On the right side, we'll multiply 2 by both 5c and 18. This gives us:

• $4 * 4c = 16c$
• $4 * -3 = -12$
• $2 * 5c = 10c$
• $2 * 18 = 36$

Putting it all together, our equation becomes:

$16c - 12 = 10c + 36$

See? We've already made the equation look a lot simpler. In this step, we've carefully applied the distributive property to both sides of the equation. We multiplied the terms outside the parentheses by each term inside. This expansion is a key step towards simplifying the equation and getting closer to solving for 'c'. Remember, the distributive property is crucial for equations containing parentheses because it lets us remove them and simplify the expression. Be super careful with the signs (positive and negative) when multiplying. A small mistake here can change the whole answer.

Step 2: Combine Like Terms

Now, let's get all the 'c' terms on one side of the equation and the constant terms (the numbers without 'c') on the other side. A common strategy is to move the terms with 'c' to the left side. To do this, we'll subtract 10c from both sides of the equation. This keeps the equation balanced. The equation looks like this:

$16c - 12 - 10c = 10c + 36 - 10c$

This simplifies to:

$6c - 12 = 36$

Great job! Now we'll isolate the constant terms to the right side of the equation. We add 12 to both sides of the equation to keep it balanced. This gives us:

$6c - 12 + 12 = 36 + 12$

Which simplifies to:

$6c = 48$

This step is all about organizing the equation to make it easier to solve. We've used subtraction and addition to group similar elements. Combining like terms is a fundamental algebraic technique that simplifies equations. It makes the next step, where we isolate the variable, much easier. Note that when you move terms from one side of the equation to another, you must change their signs (from positive to negative or vice versa).

Step 3: Isolate the Variable

We're almost there! Now that we have $6c = 48$, we need to isolate 'c'. To do this, we divide both sides of the equation by 6. This is because 6 is being multiplied by 'c', and division is the inverse operation of multiplication.

So, we have:

$6c / 6 = 48 / 6$

This simplifies to:

$c = 8$

And there you have it! We've found the value of 'c'. It means that the variable 'c' is equal to 8. This step involves a fundamental algebraic principle: the inverse operation. By dividing both sides of the equation by 6, we 'undo' the multiplication and isolate the variable. The isolation of the variable is the ultimate goal when solving any equation. It provides a numerical value for the unknown, resolving the problem. By dividing both sides by 6, we ensure that the equation remains balanced and the solution is correct.

Verifying the Solution: Is Our Answer Correct?

It's always a good idea to check your answer to make sure you didn't make any mistakes along the way. To check our solution, we'll substitute the value of 'c' (which is 8) back into the original equation $4(4c-3)=2(5c+18)$. If the equation holds true, then our answer is correct. Let's do it!

Original equation: $4(4c-3)=2(5c+18)$ Substitute c = 8: $4(4*8-3)=2(5*8+18)$

Now, let's simplify:

• $4 * 8 = 32$, so $4(32 - 3) = 2(40 + 18)$
• $32 - 3 = 29$, so $4 * 29 = 2(58)$
• $4 * 29 = 116$ and $2 * 58 = 116$

So, we have $116 = 116$. The equation balances! This confirms that our solution, c = 8, is correct. The verification process is a crucial step in ensuring that the solution satisfies the initial equation. It validates the correctness of the steps taken and confirms that we have successfully solved the equation. Always take a moment to double-check your work, particularly in more complex problems. Substituting the solution back into the original equation is a reliable way to catch any errors and increase your confidence in your answer. This step is about reinforcing the principle of equality and demonstrating that the value of the variable truly satisfies the original conditions. The verification confirms that c=8 is the correct solution. Remember to always check your answers to catch any potential errors and ensure the solution's accuracy. This reinforces understanding and builds confidence in solving mathematical equations.

Common Mistakes and How to Avoid Them

Even seasoned math enthusiasts sometimes stumble. It's perfectly okay! Recognizing common mistakes is a great way to improve. Let's talk about some traps and how to dodge them when solving equations like $4(4c-3)=2(5c+18)$. We'll focus on the frequent errors and how to steer clear of them. Recognizing the frequent errors will help you improve your skills and accuracy. Here's a look at common mistakes:

Mistake 1: Incorrect Distribution

A common blunder is misapplying the distributive property. For instance, forgetting to multiply each term inside the parentheses by the number outside. Always ensure that every term within the parenthesis is multiplied by the external factor. Remember to multiply each term within the parentheses by the number outside. For example, in 4(4c - 3), both 4c and -3 should be multiplied by 4, resulting in 16c - 12. Failing to do so is a frequent error. Double-check your distribution to catch these mistakes. Go through each term carefully to ensure correct multiplication.

Mistake 2: Sign Errors

Signs are sneaky, and sign errors are common, especially when subtracting negative numbers or distributing negative signs. For instance, subtracting a negative number is equivalent to adding. Always pay very close attention to the signs (+ or -) throughout the equation. Ensure that you correctly handle these when combining terms. Carefully manage each negative sign to avoid any mistakes. When multiplying terms, ensure you multiply both the number and the sign. Using parentheses and color-coding can often help in these situations. Review the signs when simplifying to prevent any potential errors.

Mistake 3: Forgetting Order of Operations

We talked about PEMDAS earlier, but it's worth restating its significance. Ignoring the order of operations can lead to mistakes. Always follow the PEMDAS order (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Always ensure correct execution with PEMDAS to avoid errors. The order of operations ensures that each mathematical operation is carried out correctly. Double-check your steps. Always ensure that the order is precisely followed to avoid errors. Following PEMDAS is crucial for arriving at the correct answer. The order of operations, as outlined by PEMDAS, is paramount for ensuring each calculation is performed correctly, from exponents to the addition, guaranteeing the final solution's accuracy.

Practice Makes Perfect: More Equations to Try

Alright, guys, you've got the basics down! Now, let's practice with some similar equations to cement your skills. Here are a few more equations for you to try solving on your own. Try these equations for some extra practice. Solving more equations allows you to reinforce your understanding and gain confidence in solving them. Work through the equations to build your skills.

1. $3(2x + 1) = 5(x - 2)$
2. $2(5y - 4) = 4(2y + 3)$
3. $6(z + 2) - 8 = 4z + 10$

Remember to follow the steps we discussed: distribute, combine like terms, isolate the variable, and verify your answer. The more you practice, the easier it will become. Don't worry if you get stuck; just go back to the steps we covered, and you'll get it. Regularly working through different problems enhances problem-solving capabilities. Each equation you solve is a step forward, making you more confident in your abilities. Remember to review the steps from this guide and apply them to each problem. Remember to review the steps from this guide and apply them to each problem.

Conclusion: You've Got This!

Fantastic job, everyone! We've successfully navigated through solving the equation $4(4c-3)=2(5c+18)$ together. You should now have a strong grasp of the process and be able to tackle similar equations with confidence. Remember to break down the problem into manageable steps, use the distributive property, combine like terms, isolate the variable, and always check your solution. Keep practicing, and you'll become a pro in no time! Keep practicing, and you'll become a pro in no time. Mathematics is a skill that improves with practice, so keep practicing. With practice, these steps will become second nature, and you'll feel more confident with solving equations. Math is a skill that improves with practice and persistence. Keep practicing and keep up the great work. Always remember to double-check your work, and don't hesitate to seek help if you need it. You've got this!