Solving 5(5x−4)=3(5x+3) A Step-by-Step Guide
Hey everyone! Let's dive into solving this equation together: 5(5x−4)=3(5x+3). If you're scratching your head, don't worry! We're going to break it down step by step, so it's super easy to follow. Math can seem daunting, but with the right approach, it becomes a puzzle that's fun to solve. This particular equation involves distributing, simplifying, and isolating the variable, which are all fundamental skills in algebra. Understanding these steps will not only help you solve this equation but also equip you to tackle more complex problems in the future. So, grab your pencil and paper, and let's get started on this mathematical adventure!
1. Understanding the Equation
First, let's make sure we all understand what the equation 5(5x−4)=3(5x+3) actually means. At its heart, an equation is a statement that two expressions are equal. In this case, we have an expression on the left side, 5(5x−4), and an expression on the right side, 3(5x+3). Our goal is to find the value of 'x' that makes both sides equal. The 'x' is called a variable, and it represents an unknown number that we need to figure out. The parentheses in the equation indicate that we'll need to use the distributive property, which is a key concept in algebra. Essentially, it means we need to multiply the number outside the parentheses by each term inside the parentheses. This is a crucial step in simplifying the equation and bringing us closer to the solution. Before we jump into the calculations, it’s good to have a clear understanding of what each part of the equation represents and the overall goal we're trying to achieve. With that foundation in place, we can move on to the next step: applying the distributive property.
2. Applying the Distributive Property
The distributive property is our best friend when we have parentheses in an equation. It allows us to simplify each side by multiplying the term outside the parentheses by each term inside. Let's start with the left side of the equation: 5(5x−4). We need to multiply 5 by both 5x and -4. So, 5 * 5x equals 25x, and 5 * -4 equals -20. That means the left side simplifies to 25x−20. Now, let's tackle the right side: 3(5x+3). Similarly, we multiply 3 by both 5x and 3. So, 3 * 5x equals 15x, and 3 * 3 equals 9. Therefore, the right side simplifies to 15x+9. Our equation now looks like this: 25x−20=15x+9. See how much simpler it looks already? By applying the distributive property, we've eliminated the parentheses and made the equation easier to work with. We're one step closer to isolating 'x' and finding its value. Next up, we'll want to gather like terms, which means getting all the 'x' terms on one side and all the constant terms on the other. This will further simplify the equation and bring us even closer to the final solution. So, keep your pencil ready, and let's move on to the next step!
3. Gathering Like Terms
Now that we've simplified both sides of the equation using the distributive property, it's time to gather like terms. Remember, our equation is currently: 25x−20=15x+9. The goal here is to get all the terms with 'x' on one side of the equation and all the constant terms (the numbers without 'x') on the other side. This makes it easier to isolate 'x' and solve for its value. Let's start by moving the 15x term from the right side to the left side. To do this, we subtract 15x from both sides of the equation. This is important because whatever we do to one side, we must do to the other to keep the equation balanced. So, 25x−15x−20=15x−15x+9 simplifies to 10x−20=9. Great! We've got all the 'x' terms on the left side. Now, let's move the constant term, -20, from the left side to the right side. To do this, we add 20 to both sides of the equation: 10x−20+20=9+20. This simplifies to 10x=29. We're getting closer! We've successfully gathered the like terms, and our equation is now in a much simpler form. We have 10x on one side and 29 on the other. The final step is to isolate 'x' completely, which we'll do by dividing both sides by the coefficient of 'x'. Keep going, you're doing awesome!
4. Isolating the Variable
We're in the home stretch! Our equation currently looks like this: 10x=29. To isolate 'x', we need to get it all by itself on one side of the equation. Since 'x' is being multiplied by 10, we need to do the opposite operation, which is division. We'll divide both sides of the equation by 10. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, (10x)/10 = 29/10. On the left side, the 10s cancel out, leaving us with just 'x'. On the right side, we have 29/10, which is a fraction. We can leave the answer as a fraction or convert it to a decimal. As a fraction, our solution is x=29/10. If we want to express it as a decimal, we simply divide 29 by 10, which gives us 2.9. So, our solution in decimal form is x=2.9. Congratulations! You've successfully isolated the variable and found the value of 'x' that makes the original equation true. We've gone through the steps of distributing, simplifying, gathering like terms, and finally, isolating the variable. Now, let's do one more crucial step to make sure our answer is correct.
5. Checking the Solution
It's always a good idea to check your solution to make sure it's correct. This helps prevent errors and gives you confidence in your answer. We found that x=2.9 (or 29/10) is the solution to the equation 5(5x−4)=3(5x+3). To check this, we'll substitute 2.9 for 'x' in the original equation and see if both sides are equal. Let's start with the left side: 5(5(2.9)−4). First, we calculate 5 * 2.9, which equals 14.5. Then we subtract 4, giving us 10.5. Finally, we multiply by 5, which results in 52.5. So, the left side of the equation equals 52.5 when x=2.9. Now, let's check the right side: 3(5(2.9)+3). Again, we start with 5 * 2.9, which is 14.5. Then we add 3, giving us 17.5. Finally, we multiply by 3, which also results in 52.5. So, the right side of the equation equals 52.5 when x=2.9. Since both sides of the equation are equal when x=2.9, we can confidently say that our solution is correct! We've successfully solved the equation and verified our answer. You've done a fantastic job following along and working through each step. Solving equations like this is a fundamental skill in algebra, and you've now added another tool to your math toolbox. Keep practicing, and you'll become a master equation solver in no time!
Conclusion
Alright, guys! We've successfully navigated the equation 5(5x−4)=3(5x+3) together. We started by understanding the equation, then we applied the distributive property, gathered like terms, isolated the variable, and finally, checked our solution. Each step was crucial in arriving at the correct answer: x=2.9. This process highlights the importance of following a systematic approach when solving algebraic equations. By breaking down the problem into smaller, manageable steps, even complex-looking equations become solvable. Remember, math is like building with blocks; each concept builds upon the previous one. Mastering these fundamental skills, like the ones we used today, will pave the way for tackling more advanced math topics in the future. So, don't be afraid to practice and challenge yourself. The more you practice, the more confident you'll become in your mathematical abilities. And always remember, there's a solution to every equation – you just need to find the right path to get there. Keep up the great work, and I'll catch you in the next math adventure! Happy solving!
