Unveiling The Equation With No Solution: A Mathematical Mystery

Hey math enthusiasts! Let's dive into a fun little puzzle that Felix cooked up. He's got a bunch of equations, but there's a catch: only one of them is a total head-scratcher with no solution. Our mission? To crack the code and figure out which equation is the unsolvable rebel. This is a classic algebra problem that tests your skills in simplifying and solving equations. So, grab your pencils, and let's get started. We'll break down each equation step-by-step, making sure we understand how to identify when an equation just won't cooperate. Ready to become equation detectives?

Diving into the Equations: A Step-by-Step Breakdown

Alright, guys, let's roll up our sleeves and take a look at the equations Felix gave us. We're going to examine each one individually, like we're detectives examining clues at a crime scene. Our goal is to simplify each equation and see if we can find a value for 'x' that makes it all true. Remember, the equation with no solution is the one that leads to a contradiction, something that just isn't possible. Let's start with the first equation: 3(x - 2) + x = 4x + 6. When it comes to solving equations, you need to follow a standard procedure. First, simplify the expressions on both sides. This usually involves getting rid of any parentheses by multiplying out, and combining like terms. Then, you want to isolate the variable, 'x', on one side of the equation. This means doing operations (adding, subtracting, multiplying, dividing) to both sides until 'x' stands alone. Finally, you can determine the solution or identify whether the equation has no solution. For the first equation, we start by distributing the 3: 3x - 6 + x = 4x + 6. Combine the 'x' terms: 4x - 6 = 4x + 6. Now, let's try to get all the 'x' terms on one side. Subtract 4x from both sides: -6 = 6. Uh oh! We've got a problem. This statement is never true. -6 can never equal 6, so this equation has no solution. We will check the other equations to make sure, but the first one is the answer. This is because the equation simplifies to a false statement, which means there is no value of x that can make the equation true.

Equation 2: 3(x - 2) + x = 4x - 6

Let's move on to equation two: 3(x - 2) + x = 4x - 6. Following the same steps, we first distribute the 3: 3x - 6 + x = 4x - 6. Combining like terms on the left side gives us: 4x - 6 = 4x - 6. Now, let's try to isolate 'x'. Subtract 4x from both sides: -6 = -6. This is a true statement, but it doesn't give us a specific value for x. Instead, it tells us that the equation is true for all values of x. This type of equation is called an identity, not an equation with no solution. So, equation two has infinitely many solutions, not none. An equation having infinitely many solutions happens when both sides of the equation are essentially the same. For example, if you have x + 2 = x + 2, any value you plug in for x will make the equation true. In our case, the equation simplifies to -6 = -6, which is always true, no matter what x is.

Equation 3: 3(x - 2) + x = 2x - 6

Alright, let's give the third equation a go: 3(x - 2) + x = 2x - 6. Let's distribute the 3: 3x - 6 + x = 2x - 6. Combining the 'x' terms: 4x - 6 = 2x - 6. Now, to isolate 'x', subtract 2x from both sides: 2x - 6 = -6. Add 6 to both sides: 2x = 0. Finally, divide both sides by 2: x = 0. So, this equation has a solution: x = 0. This means we have a specific value for x that makes this equation true, therefore, this isn't our unsolvable equation. Always remember to check your work; plugging x = 0 back into the original equation will confirm that our solution is indeed correct. Now, by substituting x = 0, we get 3(0 - 2) + 0 = 2(0) - 6, which simplifies to -6 = -6. This confirms that x = 0 is a correct solution for this equation. Solving linear equations systematically, involving distribution, combining like terms, isolating the variable, and verifying the solution, is an important skill in math.

Equation 4: 3(x - 2) + x = 3x - 3

Finally, let's tackle equation four: 3(x - 2) + x = 3x - 3. Distribute the 3: 3x - 6 + x = 3x - 3. Combine the 'x' terms: 4x - 6 = 3x - 3. Now, let's isolate 'x'. Subtract 3x from both sides: x - 6 = -3. Add 6 to both sides: x = 3. This equation has a solution too: x = 3. This means that we do have a valid solution for the equation. So, out of all these equations, it has a solution and therefore cannot be the equation with no solution. We can confirm this is correct by substituting x = 3, which gives us 3(3 - 2) + 3 = 3(3) - 3. This simplifies to 6 = 6. This confirms that x = 3 is the correct solution for this equation. This is the last equation in our list, and we've successfully found a solution in it as well. It's a great demonstration of the importance of precise algebraic manipulations and the ability to distinguish between equations with one solution, infinite solutions, and no solutions.

The Unsolvable Equation Revealed

So, after all that detective work, we've found our answer. The equation that has no solution is the first one: 3(x - 2) + x = 4x + 6. When we simplified it, we ended up with the contradictory statement -6 = 6, which can never be true. This tells us that there's no value of x that can make this equation balance. The key takeaway here is to always be on the lookout for those contradictions. When solving equations, if you end up with a statement that's obviously false (like 5 = 7 or -2 = 0), then you know the equation has no solution. The steps for solving linear equations are crucial, but understanding the possible outcomes—one solution, infinite solutions, or no solution—is just as important. Knowing this allows you to quickly recognize the nature of the equation and avoid unnecessary calculations. The next time you're faced with an equation, remember Felix's puzzle, and apply your newfound skills! You've got this, guys!