Solve For X: -5(x-20)=35 - Step-by-Step Solution

Hey everyone! Let's dive into solving a fun little equation today. We've got this equation: $-5(x-20) = 35$, and our mission is to find out which value of $x$ makes it true. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's quickly understand what the equation is telling us. We have $-5$ multiplied by the expression $(x - 20)$, and the result equals $35$. Our goal is to isolate $x$ on one side of the equation so we can see what value it needs to be. Equations like this are fundamental in algebra, and mastering them opens the door to more complex mathematical concepts. They're like the building blocks of problem-solving, showing up in various fields from engineering to economics. So, understanding these basics is super crucial for anyone looking to boost their math skills.

When we look at $-5(x-20) = 35$, we're essentially saying that if we take a number (which we're calling $x$), subtract 20 from it, and then multiply the result by -5, we should end up with 35. It's like a little puzzle, and our job is to find the missing piece – the value of $x$. To do this, we'll use some basic algebraic techniques that will help us unravel the equation and get to the solution. Think of it as reverse-engineering a problem to find the answer, a skill that’s not just useful in math but in everyday life too! So, let's roll up our sleeves and start solving this puzzle together.

Step-by-Step Solution

Step 1: Distribute the -5

Okay, first things first, we need to get rid of those parentheses. To do that, we'll distribute the $-5$ across the terms inside the parentheses. This means we'll multiply $-5$ by both $x$ and $-20$. Remember the distributive property? It's a key tool in algebra, allowing us to simplify expressions and equations. When we distribute, we're essentially applying the multiplication to each term inside the parentheses, ensuring that we account for everything. It's like making sure everyone gets a fair share in a group – each term gets its due multiplication!

So, let's break it down: $-5 * x = -5x$ and $-5 * -20 = 100$. Notice that when we multiply two negative numbers, we get a positive number. This is a crucial rule to remember in math, and it's super helpful in avoiding mistakes. With that in mind, our equation now looks like this: $-5x + 100 = 35$. See? We've already made progress by simplifying the equation. The parentheses are gone, and we're one step closer to isolating $x$. This is how we chip away at a problem, making it more manageable and bringing the solution into clearer view. Let's keep going!

Step 2: Isolate the Term with x

Alright, now we need to get the term with $x$ (which is $-5x$) all by itself on one side of the equation. To do this, we'll subtract $100$ from both sides. Remember, whatever we do to one side of the equation, we have to 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 side to keep it level. This principle is fundamental in solving equations, ensuring that we maintain equality throughout the process.

So, let's do it: $-5x + 100 - 100 = 35 - 100$. This simplifies to $-5x = -65$. Notice how the $100$ on the left side cancels out, leaving us with just the term we want to isolate. We're making good progress! By isolating the term with $x$, we're setting ourselves up for the final step, where we'll actually find the value of $x$. This is all about strategic manipulation, using mathematical operations to gradually reveal the answer. Keep it up, guys!

Step 3: Solve for x

Okay, we're in the home stretch now! We have $-5x = -65$, and we want to find $x$. To do this, we'll divide both sides of the equation by $-5$. Again, it's all about keeping the equation balanced – what we do to one side, we do to the other. Division is the inverse operation of multiplication, so it's the perfect tool to undo the multiplication by $-5$ that's currently attached to $x$. This is a common technique in algebra, and it's super effective in isolating variables and finding their values.

Let's divide: $(-5x) / -5 = -65 / -5$. This simplifies to $x = 13$. And there we have it! We've found the value of $x$ that makes the equation true. Notice that we divided a negative number by a negative number, which resulted in a positive number. This is another key rule to keep in mind when working with negative numbers. Solving for $x$ involves carefully applying inverse operations, step by step, until we isolate the variable. It's a process of methodical deduction, and it's incredibly satisfying when you arrive at the correct answer.

Checking the Answer

Before we celebrate too much, it's always a good idea to check our answer. This helps us make sure we didn't make any mistakes along the way. To check our answer, we'll substitute $x = 13$ back into the original equation and see if it holds true. Checking our work is a critical step in problem-solving, not just in math but in any field. It's about ensuring accuracy and building confidence in our solutions. Think of it as proofreading a document before submitting it – you want to catch any errors and make sure everything is correct.

So, let's plug it in: $-5(13 - 20) = 35$. First, we simplify the expression inside the parentheses: $13 - 20 = -7$. Now we have $-5 * -7 = 35$. And guess what? $-5 * -7$ does indeed equal $35$. Our answer checks out! This confirms that $x = 13$ is the correct solution. Checking your work not only validates your answer but also reinforces your understanding of the problem-solving process. It's a great habit to develop, and it will serve you well in all your mathematical endeavors.

The Correct Answer

So, after all that solving and checking, we've found that the value of $x$ that makes the equation $-5(x - 20) = 35$ true is $13$. Looking back at our options, the correct answer is A. 13. Awesome job, guys! We tackled the equation step-by-step, making sure we understood each move along the way. Remember, problem-solving in math is like building a puzzle – each step fits together to reveal the final solution. And by checking our work, we've ensured that our puzzle is complete and correct.

This was a great exercise in applying basic algebraic principles, like the distributive property and the importance of keeping equations balanced. These skills are fundamental in math, and they'll be super helpful as you tackle more complex problems. So, keep practicing, stay curious, and remember that every equation is just a puzzle waiting to be solved. You've got this!

Why Other Options Are Incorrect

It's also beneficial to understand why the other options are incorrect. This not only reinforces the correct solution but also helps in avoiding common mistakes in the future. Let's quickly examine why options B, C, and D are not the correct answers.

B. -3

If we substitute $x = -3$ into the equation, we get: $-5(-3 - 20) = -5(-23) = 115$, which is not equal to $35$. So, $-3$ is not the correct value for $x$. This mistake might arise from incorrectly handling the negative signs or misunderstanding the order of operations. It's a good reminder to pay close attention to the signs and follow the correct sequence of steps when simplifying expressions.

C. -11

Substituting $x = -11$ gives us: $-5(-11 - 20) = -5(-31) = 155$, which is also not equal to $35$. So, $-11$ is not the correct answer either. This error could stem from a similar mistake as in option B, such as mishandling negative signs or incorrect distribution. It highlights the importance of carefully reviewing each step to avoid such errors.

D. 27

Finally, let's try $x = 27$: $-5(27 - 20) = -5(7) = -35$, which is not equal to $35$. This option is close, but the sign is incorrect. This mistake could be due to overlooking the negative sign in front of the $5$ or making a simple arithmetic error. It emphasizes the need for meticulous calculation and attention to detail.

By understanding why these options are incorrect, we gain a deeper insight into the problem-solving process and the importance of accuracy in each step. It's not just about finding the right answer but also about understanding the potential pitfalls and how to avoid them. So, keep analyzing, keep learning, and you'll become a math whiz in no time!

Conclusion

We've successfully navigated through the equation $-5(x - 20) = 35$ and found that the correct value of $x$ is $13$. We did this by following a clear, step-by-step approach: distributing, isolating the variable term, and finally solving for $x$. We also checked our answer to make sure it was correct, and we explored why the other options were not the solution. This process not only helps us solve this specific problem but also equips us with valuable skills for tackling other mathematical challenges.

Remember, math is like a journey – each problem is a new destination, and the skills we learn along the way are our tools. By practicing and understanding the fundamental principles, we can confidently approach any equation or problem that comes our way. So, keep exploring, keep learning, and most importantly, keep enjoying the process. You're doing great, guys! Let's continue to build our math skills together and conquer the world of equations!