Solving Equations: How Many Solutions?

Hey everyone! Ever stumble upon an equation and wonder, "How many answers could this even have?" Well, today, we're diving deep into the world of linear equations and figuring out exactly that. We'll be tackling the equation: 0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20). Our mission? To uncover whether it has zero, one, two, or infinitely many solutions. Buckle up, because we're about to transform from equation-solving rookies to solution-detecting pros! Keep reading, it will be fun, I promise. This question is a classic in the realm of algebra, designed to test your understanding of how equations behave and what their solutions represent. It's not just about crunching numbers; it's about seeing the bigger picture and understanding what the result tells you about the equation itself. So, are you ready to learn about how to find the number of solutions? Let's get started!

Decoding the Equation's Secrets

Alright, first things first, let's break down the given equation: 0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20). This is a linear equation, meaning that when graphed, it would form a straight line. The key to solving this, and finding out the number of solutions, lies in simplifying and isolating x. To start, we'll use the distributive property to get rid of those parentheses. For the left side of the equation, we multiply 0.75 by both x and 40. This gives us 0.75x + 30. On the right side, we'll do the same: 0.35 * x + 0.35 * 20 + 0.35 * x + 0.35 * 20, which simplifies to 0.35x + 7 + 0.35x + 7. Now, let's combine like terms on the right side to get 0.70x + 14. Thus, our equation becomes 0.75x + 30 = 0.70x + 14. Here's where the real magic begins. Our main goal is to isolate x to determine the solution. We will subtract 0.70x from both sides to get 0.05x + 30 = 14. After that, we subtract 30 from both sides, which gives us 0.05x = -16. To finally isolate x, we divide both sides by 0.05, and that gives us x = -320. Now we have a clear value for x, which means... we have ONE solution. This process of simplifying and isolating the variable is a fundamental skill in algebra. The ability to manipulate and transform equations is crucial for not only solving them but also for understanding their underlying structure. Always remember to check your work! Substituting -320 back into the original equation will confirm whether our solution is correct. If the equation holds true, then our solution is spot-on. If it's false, we will have to go back and check our steps for any errors. Understanding each step, like the distributive property or combining like terms, will enable you to solve the equation. So, keep practicing, and equations will be no match for you!

Unveiling the Solution Types

Now that we have gone over the equation, it's a great time to talk about the types of solutions you might encounter when dealing with linear equations. A linear equation, at its core, describes a straight line. The number of solutions directly relates to the intersections of lines. So, let's break down the scenarios that could occur. First, we have the case where there is one solution, as we found in our example. This happens when the equation simplifies down to a unique value for x. This means that the graph of the equation would be a single point, where the lines intersect. Next, we have no solution, meaning there is no value of x that will satisfy the equation. This happens when, after simplifying, we get a false statement, like 2 = 5. Graphically, this represents parallel lines that never intersect. Finally, we have the case with infinitely many solutions. This occurs when, after simplifying, we get a true statement, such as 3 = 3. In this scenario, the equation is essentially an identity, and any value of x will make it true. Graphically, this represents the same line, where one line is perfectly overlapping the other. Understanding these types will make you an equation solver. This framework of understanding solution types doesn't just apply to linear equations, but also provides a conceptual base for understanding solutions in more complex mathematical scenarios. So, as you advance in your mathematical journey, remember these concepts, and you will be well-equipped to tackle more intricate equations and problems. Keep learning, and you'll find that equations can reveal much about the world around us. With practice, you'll start to see patterns and develop a knack for predicting the type of solution an equation will have even before you solve it!

Mastering the Equation: Step-by-Step

Let's recap how to find the number of solutions to this equation. Understanding the steps will help you out for future problems! First, always start by simplifying both sides of the equation. Use the distributive property to clear any parentheses, and combine like terms. If you have any terms with x on both sides, bring them to one side by adding or subtracting. This is the stage where you want to isolate x. After that, isolate x. Perform the necessary operations (addition, subtraction, multiplication, or division) to get x alone on one side of the equation. Ensure that you perform the same operations on both sides to maintain the equation's balance. Once x is isolated, you'll have a value for x. This value is your solution. In this case, it was -320. At the end, you should check your work. Substitute your solution back into the original equation to verify that it's correct. If the equation holds true, then your solution is accurate, and it tells you that there's one solution. If you arrive at an equation that is always true, such as 2 = 2, then the equation has infinitely many solutions. This means that any value of x will satisfy the equation. If you arrive at an equation that is never true, such as 2 = 5, then the equation has no solution. This indicates that there's no value of x that will satisfy the equation. By following these steps, you can confidently determine the number of solutions to any linear equation. Remember, practice is key. The more you work with equations, the more familiar you will become with recognizing patterns and understanding the different types of solutions. Don't be discouraged if you don't get it immediately. Keep practicing, and you'll become a pro at equation solving. You will soon master the art of solving equations! Don't be afraid to take your time and break down each step. Before you know it, you'll be able to solve these equations without any problems. So, keep up the great work!

Conclusion: Finding the Right Answer!

Alright, guys! We've made it to the end. So, what's the answer to our original equation? Let's refresh our minds: the equation 0.75(x + 40) = 0.35(x + 20) + 0.35(x + 20) simplifies to a single solution. This process shows that the equation has only one solution, which is x = -320. The journey of solving this equation wasn't just about getting an answer; it was a lesson in understanding how equations work. You learned about simplifying, isolating variables, and, most importantly, identifying the number of possible solutions. Remember, math isn't just about memorizing formulas; it's about understanding the concepts behind them. In the end, equations are an awesome tool that will help you solve real-world problems. Whether you're a student, professional, or simply someone who loves a good challenge, the ability to solve equations is a valuable skill. If you keep practicing, then you will succeed. So, keep up the great work! And now you know how to determine the number of solutions for a linear equation.