Equation A And Equation B Analysis Determining The True Statement
Hey guys! Let's dive into the fascinating world of equations and figure out what makes them tick. We've got two equations here Equation A and Equation B and we need to determine which statements about their solutions are actually true. Buckle up, because we're about to embark on a mathematical adventure!
Cracking the Code of Equation A
Equation A: $3(2x - 5) = 6x - 15$
So, the main keyword here is Equation A, and it's all about understanding its solutions. When we say solutions, we're talking about the values of 'x' that make the equation true. Let's break it down step by step. First, we need to simplify the equation by distributing the '3' on the left side. This means multiplying both terms inside the parentheses by 3:
This simplifies to:
Now, take a good look at this equation. Notice anything interesting? The left side is exactly the same as the right side! This is a huge clue. What it means is that no matter what value we plug in for 'x', the equation will always be true. Let's try a few examples just to be sure.
If we let x = 0:
-15 = -15$ (True!) If we let x = 1: $6(1) - 15 = 6(1) - 15
-9 = -9$ (Also true!) If we let x = -1: $6(-1) - 15 = 6(-1) - 15
-21 = -21$ (Still true!) You see, *no matter what number we substitute for x*, the equation holds true. This is because Equation A is an **identity**. An identity is an equation that is always true, regardless of the value of the variable. So, what does this mean for the number of solutions? Well, since any value of x works, there are an infinite number of solutions. Think of it like this: you could keep plugging in numbers forever, and you'd never find one that doesn't work. The concept of **infinite solutions** is crucial here. It's not just that there are a lot of solutions; there are literally an endless number of them. This is a special case in algebra, and it's important to recognize when it occurs. When you encounter an equation where both sides are identical after simplification, you're dealing with an identity, and that means infinite solutions. In summary, Equation A is an identity, and identities have an infinite number of solutions. We've shown this by simplifying the equation and by testing different values of x. No matter what we try, the equation remains true. This understanding of identities is fundamental in mathematics, so make sure you've got a solid grasp on it! ## Unraveling the Mystery of Equation B **Equation B: $2 + 3x = 3x - 4$** Now, let's turn our attention to Equation B and see what secrets it holds. The main focus here is understanding the solutions, if any, that satisfy this equation. Our approach will be similar to what we did with Equation A we'll try to simplify the equation and see what we end up with. The goal is to isolate 'x' on one side of the equation, but let's see what happens. First, we want to get all the 'x' terms on one side. A common strategy is to subtract the smaller 'x' term from both sides. In this case, the 'x' terms are the same (3x), so we can subtract 3x from both sides: $2 + 3x - 3x = 3x - 4 - 3x
This simplifies to:
Whoa, hold on a second! This looks a bit strange, doesn't it? We've ended up with the statement 2 = -4. This is definitely not true! 2 is not equal to -4. So, what does this mean for the solutions of Equation B? It means that there are no values of x that can make this equation true. No matter what number we plug in for 'x', the equation will always be false.
This is another special case in algebra, and it's just as important to recognize as the case of infinite solutions. When you simplify an equation and end up with a false statement (like 2 = -4), it means the equation has no solution. There's no value of 'x' that can satisfy the equation. Think of it like trying to find a key that opens a lock, but the lock is designed in a way that no key will ever fit. The equation is inherently contradictory.
To further illustrate this, let's try plugging in a couple of values for 'x' just to see what happens.
If we let x = 0:
2 = -4$ (False!) If we let x = 1: $2 + 3(1) = 3(1) - 4
5 = -1$ (Also false!) As you can see, no matter what value we try, the equation remains false. This confirms that Equation B has no solution. The key takeaway here is that when simplifying an equation leads to a contradiction, you're dealing with an equation that has no solution. This is a fundamental concept in algebra, and understanding it will help you solve a wide range of problems. In conclusion, Equation B is a contradictory equation, and contradictory equations have no solutions. We've demonstrated this by simplifying the equation and by testing different values of x. No matter what we try, the equation remains false. ## Comparing Equation A and Equation B The Final Verdict Alright, guys, we've dissected both Equation A and Equation B. Now, let's put the pieces together and see which statements about them are true. We've established that: * Equation A has an infinite number of solutions because it's an identity. * Equation B has no solution because it's a contradictory equation. Now, let's look at the original options and see which one aligns with our findings: A. Equation A and Equation B have an infinite number of solutions. B. Equation A has an infinite number of solutions, and Equation B has no solution. Based on our analysis, option B is the correct answer. Equation A indeed has an infinite number of solutions, and Equation B has no solution. This exercise highlights the importance of simplifying equations and understanding the different types of solutions that can arise. Equations can have one solution, infinite solutions, or no solution at all. Being able to identify these cases is a crucial skill in algebra and beyond. Remember, when an equation simplifies to an identity (a true statement), it has infinite solutions. And when it simplifies to a contradiction (a false statement), it has no solution. By carefully analyzing each equation, we were able to determine the nature of their solutions and arrive at the correct answer. This is the power of mathematics it allows us to solve problems by breaking them down into smaller, manageable steps. So, there you have it! We've successfully navigated the world of equations and solutions. Keep practicing, and you'll become a master of mathematical problem-solving in no time!