Mastering Linear Equations: Solving 7x - 9 = 5x + 3

Hey Guys, Let's Tackle This Equation Together!

Alright, math enthusiasts and problem-solvers, listen up! We've all been there, right? Staring at an equation, doing the math, getting an answer, and then... boom! The verification just doesn't add up. It's like your calculator is playing a cruel joke, and you're left scratching your head, wondering what cosmic error you possibly made. Well, you're absolutely not alone, and today we're going to dive headfirst into one of those tricky situations, specifically with the equation 7x - 9 = 5x + 3. Many of you, like our friend who posted this query, might get x = 12 when solving, only to find it goes completely sideways during the verification process. Don't sweat it, because by the end of this deep dive, you'll not only understand why x = 12 isn't the right answer but also master the steps to correctly solve and confidently verify this (and many other!) linear equations.

Linear equations are the bread and butter of algebra, forming the foundational bedrock for more complex mathematical concepts. They're everywhere, from calculating your budget to designing bridges, and even in the algorithms that power your favorite social media apps. Understanding them isn't just about passing a math test; it's about developing critical thinking skills and problem-solving muscle that pays dividends in almost every aspect of life. When you encounter a linear equation like 7x - 9 = 5x + 3, the core goal is always the same: isolate the variable, which in this case is 'x'. This means getting 'x' all by itself on one side of the equals sign, leaving a number on the other side. This number, if you've done everything correctly, is your solution! The beauty of these equations, guys, is that they always have one unique solution, and there's a straightforward, step-by-step process to find it. However, tiny slips – a misplaced negative sign, an incorrect operation, or an accidental addition instead of subtraction – can throw your entire calculation off course, leading to that frustrating verification failure. So, buckle up, because we're about to demystify 7x - 9 = 5x + 3 and equip you with the knowledge to ace any similar algebraic challenge. Let's get to it!

Unpacking the Mystery: Step-by-Step Solution to 7x - 9 = 5x + 3

Step 1: Gathering Our Variables (The 'x' Crew)

Alright, guys, let's kick things off with our main objective: getting all the 'x' terms to one side of the equation. Think of it like organizing your toolbox – you want all the wrenches together, all the screwdrivers together, and so on. In our algebraic toolbox, the 'x' terms are our wrenches, and we need them on one side. Our equation, as a reminder, is 7x - 9 = 5x + 3. Notice we have 7x on the left and 5x on the right. To consolidate them, we need to move one of the 'x' terms. The easiest way to do this is to subtract the smaller 'x' term from both sides. Why subtract? Because we want to eliminate it from one side, and the opposite of adding 5x is subtracting 5x. And remember, whatever you do to one side of the equation, you must do to the other to keep it balanced, just like a seesaw!

So, starting with 7x - 9 = 5x + 3, we're going to subtract 5x from both sides. This is a super crucial step, and it's where some common mistakes sneak in. Sometimes, folks accidentally add 5x to both sides, thinking they're combining terms, but that would give you 12x on the left and 10x on the right, making things more complicated. No, no, no! Our goal is to cancel out the 5x on the right. So, let's write it out clearly:

7x - 9 - 5x = 5x + 3 - 5x

On the right side, 5x - 5x happily cancels out to zero. That's exactly what we wanted! On the left side, we combine 7x - 5x, which simplifies to 2x. Now our equation looks much cleaner:

2x - 9 = 3

See? We've successfully gathered our 'x' terms! This first move is often the make-or-break moment. By carefully subtracting 5x from both sides, we've brought ourselves one giant leap closer to isolating that elusive 'x'. It's all about precision and understanding those fundamental algebraic operations. Don't rush this step, and always double-check your signs, because a small error here can snowball into a completely wrong answer down the line. We're on a roll now, let's move on to the next piece of the puzzle!

Step 2: Isolating the 'x' (Saying Goodbye to Constants)

Alright, awesome job getting all those 'x' terms organized! Now that we have 2x - 9 = 3, our next mission, should we choose to accept it (and we definitely do!), is to start isolating the 'x'. This means we need to get rid of any plain numbers, or constants, that are hanging out on the same side as our 'x' term. In this case, we've got a -9 chilling with our 2x on the left side of the equation. To get 'x' closer to being by itself, we need to move that -9 to the other side.

How do we do that, you ask? Easy peasy! Just like before, we use the opposite operation. Since we have a -9 (or minus 9), the opposite operation is to add 9. And, just like we hammered home in Step 1, whatever you do to one side of the equation, you must do to the other to keep everything perfectly balanced. Think of it like those old-school scales: if you add weight to one side, you have to add the same weight to the other to keep it level.

So, starting with 2x - 9 = 3, we're going to add 9 to both sides. Let's write that out so it's super clear:

2x - 9 + 9 = 3 + 9

On the left side, -9 + 9 cancels out beautifully, leaving us with just 2x. Perfect! On the right side, 3 + 9 gives us 12. Look at that! Our equation has transformed into something much simpler:

2x = 12

Isn't that just satisfying? We've successfully moved all the constants to one side, and now our 'x' term is almost completely isolated. This step is another common area for small mistakes, especially with signs. If you had, say, 2x + 9 = 3, you'd need to subtract 9 from both sides. Always be mindful of whether you need to add or subtract to cancel out the constant term. Getting 2x = 12 means we are super close to finding the value of 'x'. We've done the heavy lifting of rearrangement, and now it's time for the grand finale. One more step, and we'll unveil the true identity of 'x'! You guys are doing great – keep that mathematical momentum going!

Step 3: Finding Our 'x' (The Grand Finale!)

Alright, math legends, we've reached the final frontier in solving this equation! We've successfully navigated through gathering our 'x' terms and moving our constants, and now we're left with the pristine 2x = 12. This is where we get to reveal the true value of 'x'! What 2x = 12 essentially means is "two times 'x' equals twelve." Our ultimate goal, remember, is to find out what one 'x' is equal to. So, if we have two 'x's, and their total value is 12, what do you think we need to do to find the value of a single 'x'?

Yep, you guessed it! We need to divide! Since 'x' is being multiplied by 2, the inverse (or opposite) operation to undo that multiplication is division. And, as our golden rule dictates, whatever we do to one side of the equation, we must do to the other side to keep that precious balance. No cheating, no shortcuts – just fair and square mathematical operations across the board!

So, taking our equation 2x = 12, we are going to divide both sides by 2. Let's lay it out clearly:

2x / 2 = 12 / 2

On the left side, 2x / 2 simplifies perfectly to just x. That's it! Our variable is finally isolated. Mission accomplished on that front! On the right side, 12 / 2 gives us a neat and tidy 6. And there you have it, folks! The moment of truth:

x = 6

Boom! We've found our 'x'! The correct solution to the equation 7x - 9 = 5x + 3 is x = 6. This is the one and only value of 'x' that will make both sides of the original equation equal. This final division step is often the easiest, but it's the culmination of all the careful work you put in during the previous steps. It's important to remember that 'x' represents a specific numerical value that satisfies the conditions of the equation. If at any point you get a fraction or a decimal, that's totally fine too; not all answers are clean whole numbers, but the process remains exactly the same. So, our friend who initially got x = 12 now knows what the correct answer should be. But why did x = 12 feel so wrong? That's what we're tackling next: the crucial step of verification, and why getting x = 12 can lead to some major head-scratching moments. Let's make sure we truly understand why x = 6 is the champ!

The Verification Vexation: Why Did x=12 Go Wrong?

This is where the rubber meets the road, guys! Our friend's original problem perfectly highlights a common frustration: "cuando la hago x me da 12, pero cuando quiero verificarla me da mal, no entiendo que hice mal." It’s a classic scenario, and honestly, it's why verification is such a critical final step in solving any equation. Getting an answer is one thing; making sure it actually works in the original equation is another. If your verification fails, it's a huge flashing red light telling you, "Hey, something went wrong earlier! Go back and check your work!" So, let's break down exactly why x=12 failed the verification test for our equation 7x - 9 = 5x + 3, and then, of course, we'll confirm the correct answer x=6 with a flawless verification.

Let's Check x=12 Together (and See the Problem)

Okay, so let's put x = 12 to the test in the original equation 7x - 9 = 5x + 3. This is how you verify: you take the value you think is the solution and substitute it back into every instance of 'x' in the original equation. Then, you simplify both sides of the equation independently. If your value for 'x' is correct, both sides must equal the same number. If they don't, then x is not the solution. Simple as that!

Let's substitute x = 12 into the left-hand side (LHS) of the equation first:

LHS: 7x - 9 Substitute x = 12: 7(12) - 9 Multiply: 84 - 9 Subtract: 75

So, the left side, when x = 12, evaluates to 75. Keep that number in mind! Now, let's do the same for the right-hand side (RHS):

RHS: 5x + 3 Substitute x = 12: 5(12) + 3 Multiply: 60 + 3 Add: 63

Alright, so the right side, when x = 12, evaluates to 63. Now, let's compare our results:

Is 75 = 63?

Absolutely not! 75 is clearly not equal to 63. This mismatch, guys, is the undeniable proof that x = 12 is not the correct solution to the equation 7x - 9 = 5x + 3. The verification process caught the error, which is exactly what it's designed to do! While we don't know the exact step where our friend made a mistake, common slips leading to x = 12 for this specific equation often include adding 5x to 7x instead of subtracting (which would give 12x - 9 = 3, leading to 12x = 12, and thus x = 1), or perhaps making an error with the constants, like subtracting 3 instead of adding it. Regardless of the specific error, the verification step is your ultimate safeguard. It tells you immediately if you're on the right track or if it's time to retrace your steps. Now that we've seen why x = 12 fails, let's look at a successful verification to truly understand what a correct answer looks like.

The Right Way to Verify: Using x=6

Okay, guys, now that we've seen the painful reality of a failed verification with x = 12, let's do the correct verification using the answer we found: x = 6. This is the moment of truth where we prove that x = 6 is indeed the champion solution for our equation 7x - 9 = 5x + 3. The process is identical: substitute x = 6 into both sides of the original equation and simplify each side independently. If our math is solid, both sides must yield the exact same value. This will give us that sweet, sweet satisfaction of knowing we've solved it perfectly!

Let's start by substituting x = 6 into the left-hand side (LHS) of the equation:

LHS: 7x - 9 Substitute x = 6: 7(6) - 9 First, we perform the multiplication: 42 - 9 Then, we do the subtraction: 33

So, when x = 6, the entire left side of our equation simplifies down to 33. That's our target number for the other side! Now, let's move over to the right-hand side (RHS) and do the same substitution and simplification:

RHS: 5x + 3 Substitute x = 6: 5(6) + 3 First, perform the multiplication: 30 + 3 Then, do the addition: 33

Fantastic! The right side, when x = 6, also simplifies down to 33. Now, let's bring it all together and compare our results:

Is 33 = 33?

Absolutely! Yes, 33 is perfectly equal to 33! This, my friends, is the sound of success in algebra. This equality confirms, beyond a shadow of a doubt, that x = 6 is the correct and true solution to the equation 7x - 9 = 5x + 3. When both sides match up perfectly after substituting your solution, it’s like getting a big green checkmark on your work. It signifies that every step you took to solve the equation was accurate and that your answer holds up under scrutiny. This feeling of certainty is why verification isn't just an optional extra; it's an indispensable part of the problem-solving process. It builds confidence in your mathematical abilities and helps you catch those sneaky errors before they become bigger headaches. So, remember this successful verification – it’s the goal every time you tackle a linear equation!

Pro Tips for Conquering Any Linear Equation

Alright, team, we've gone on quite the journey with 7x - 9 = 5x + 3, solving it step-by-step and understanding the absolute importance of verification. But the lessons we've learned extend far beyond just this one equation! These tips are your secret weapon for tackling any linear equation with confidence, making sure you rarely hit that frustrating