Finding Equations With One Root: A Step-by-Step Guide

Hey guys! Let's dive into some algebra and figure out which of the given equations has only one solution, or as we like to call it, one root. This is a classic problem that tests your understanding of how equations work. We'll break down each equation step by step, showing you how to find the root if it exists, or explain why some equations might have no solutions or infinitely many. Understanding this concept is super important as you level up your math skills! So, grab your pencils and let's get started. We'll go through each option (A, B, C, and D) one by one, making sure we understand what's happening in each case. The goal is to isolate 'x' on one side of the equation and see what we end up with. Ready?

Understanding Roots and Equations

Before we start, let's refresh our memory on what a root actually is. In simple terms, the root of an equation is the value of the variable (in this case, 'x') that makes the equation true. For example, if we have the equation x + 2 = 5, the root is 3 because 3 + 2 does indeed equal 5. Simple, right? But things can get a bit more interesting! Sometimes, an equation has no root (no value of 'x' makes it true), and sometimes it has infinitely many roots (every value of 'x' works). Our mission is to identify the equation that has a single, unique root. This is the cornerstone of solving linear equations, and it's a fundamental concept in algebra. We use several key principles to solve for 'x', which includes the distributive property, combining like terms, and isolating the variable. If you grasp these fundamental concepts, you'll be well-equipped to tackle more complex algebraic problems down the line.

Here’s a quick reminder of the key steps:

1. Simplify both sides: Use the distributive property and combine like terms.
2. Isolate the variable: Get all terms with 'x' on one side and constant terms on the other.
3. Solve for 'x': Divide to find the value of x.

By following these steps, we'll analyze each equation provided and reveal which one stands out with its unique solution. Knowing these steps ensures you're on the right track! This process is crucial not just for this problem but also for building a solid foundation in algebraic problem-solving. It’s like a recipe; following the steps precisely will lead you to the right answer. Stay tuned as we apply these steps to each of the equations!

Analyzing the Equations

Alright, let’s get down to the nitty-gritty and examine each equation one by one. Remember, we are looking for the equation that gives us a single, unique value for x. We'll be using the basic principles of algebra – the distributive property, combining like terms, and isolating variables – to solve each equation. This approach allows us to determine the possible values of 'x'.

Equation A: 2(x - 7) = 2x - 14

Let’s start with equation A: 2(x - 7) = 2x - 14. First, apply the distributive property to the left side: 2 * x - 2 * 7 = 2x - 14, which simplifies to 2x - 14 = 2x - 14. Now, if we try to isolate 'x', we might see something interesting happening. Let's subtract 2x from both sides: 2x - 14 - 2x = 2x - 14 - 2x. This gives us -14 = -14. Wait a minute! The 'x' disappeared, and we ended up with a statement that is always true, no matter the value of 'x'. This means that any value of 'x' will make the equation true, so it does not have a single root. This equation actually has infinitely many solutions, meaning it's not our answer.

Equation B: 2(x - 7) = 2x - 7

Next up, let's look at Equation B: 2(x - 7) = 2x - 7. Again, we start by distributing on the left side: 2 * x - 2 * 7 = 2x - 7, so we get 2x - 14 = 2x - 7. Now, let’s try to isolate ‘x’ like we did before. We subtract 2x from both sides: 2x - 14 - 2x = 2x - 7 - 2x. That simplifies to -14 = -7. This is a false statement. This means there's no value of ‘x’ that can make this equation true. This tells us that this equation has no solution, which means it doesn't have one root either. Keep going; we are getting closer.

Equation C: 2(x - 7) = 7x - 2

Okay, let's move on to Equation C: 2(x - 7) = 7x - 2. Once more, start by distributing: 2 * x - 2 * 7 = 7x - 2, or 2x - 14 = 7x - 2. Now, let’s isolate ‘x’. Subtract 2x from both sides: 2x - 14 - 2x = 7x - 2 - 2x, so we get -14 = 5x - 2. Add 2 to both sides: -14 + 2 = 5x - 2 + 2, resulting in -12 = 5x. Now, divide both sides by 5: -12 / 5 = x. This gives us x = -12/5, which is a single, valid solution! This equation gives us a unique solution, so it looks like this is our answer. The presence of a single solution confirms that this is the equation with one root. This is exactly what we were looking for. Let's make sure before we declare the answer.

Equation D: 2(x - 7) = 2x - 9

Finally, let’s check out Equation D: 2(x - 7) = 2x - 9. Distribute on the left side: 2 * x - 2 * 7 = 2x - 9, or 2x - 14 = 2x - 9. Subtract 2x from both sides: 2x - 14 - 2x = 2x - 9 - 2x. We get -14 = -9. This is another false statement, so there’s no value of 'x' that makes this equation true. This equation has no solution, which means it does not have one root.

Conclusion and Answer

So, after working through each equation, we found that only one of them has a single root. Equation C, 2(x - 7) = 7x - 2, has a unique solution: x = -12/5. This means that only Equation C has one root. All other equations either have infinitely many solutions or no solutions at all. Therefore, the correct answer is C!

Keep practicing these problems, and you'll get more comfortable with solving equations. You can use these steps for a variety of problems, and the more you practice, the easier it gets. Great job, everyone! And remember, practice makes perfect! Always remember the fundamental principles of algebra; they will guide you through more complicated problems.