Which Equation Holds True?

Hey guys! Welcome back to Plastik Magazine, where we dive deep into the cool and sometimes tricky world of math. Today, we've got a real head-scratcher for you, a problem that’s all about finding the winning ticket among a set of equations. We're going to break down each option, step-by-step, so you can see exactly how we get to the right answer. It’s not just about getting it right; it’s about understanding why it’s right, and that’s where the real fun begins in mathematics. So, grab your thinking caps, maybe a calculator if you’re feeling fancy, and let’s get started on this mathematical treasure hunt! We want to find out which of these statements accurately represents an equivalent equation after a simple distributive property step. This involves understanding basic algebra and how operations affect the equation. Let's unravel this puzzle together, piece by piece, to make sure everyone feels confident and clear on the concepts involved.

Understanding the Core Concept: Equivalent Equations

Before we jump into the nitty-gritty of each option, let's chat for a sec about what an equivalent equation actually means. In the wild world of algebra, two equations are considered equivalent if they have the exact same solutions. It's like having two different paths that lead to the same destination. In this problem, we're not being asked to solve for x just yet, but rather to check if the transformation from the first form to the second form is mathematically sound. The transformation shown in each option involves applying the distributive property. Remember that? It's where you multiply a number outside the parentheses by each term inside the parentheses. For example, in $a(b+c)$, you'd do $a*b + a*c$. We’re going to see how this applies and if the subsequent step shown maintains the equality. The key here is that any operation you perform on one side of the equation must be performed on the other side to keep it balanced. When we look at the format $k(x-c)=d$ becoming $kx=d'$, we’re checking if the distribution and the subsequent adjustment to the right side are correct. This is a fundamental skill that underpins many more complex algebraic manipulations, so getting a solid grasp on it now will pay dividends later on, trust me. We're essentially performing a single step of algebraic simplification and need to verify its accuracy.

Analyzing Option A: $4(x-5)=35$ is equivalent to $4x=40$

Alright guys, let's tackle Option A first. We've got the equation $4(x-5)=35$. The first thing we need to do is apply the distributive property. We multiply the 4 by both terms inside the parentheses: $x$ and $-5$. So, $4*x$ gives us $4x$, and $4*(-5)$ gives us $-20$. That means our equation, after distributing, becomes $4x - 20 = 35$. Now, the option claims this is equivalent to $4x = 40$. To check this, we need to see if we can get from $4x - 20 = 35$ to $4x = 40$ using valid algebraic steps. If we add 20 to both sides of $4x - 20 = 35$, we get $4x - 20 + 20 = 35 + 20$, which simplifies to $4x = 55$. Now, compare this to what the option states: $4x = 40$. Since $55$ is definitely not equal to $40$, Option A is not the winning ticket. The transformation shown is incorrect. It seems like they might have added 20 to the 35, but perhaps made a calculation error, or maybe they distributed incorrectly in the first place. Let's re-check the distribution: $4 * x = 4x$ and $4 * -5 = -20$. So, $4x - 20 = 35$. To isolate the $4x$ term, we add 20 to both sides: $4x = 35 + 20$, which gives $4x = 55$. The presented equivalence to $4x=40$ is false. This is a common type of error where a calculation mistake happens during the distribution or the subsequent isolation of a term. It’s crucial to be meticulous with each step. So, Option A is out!

Analyzing Option B: $8(x-7)=35$ is equivalent to $8x=28$

Moving on to Option B, we have $8(x-7)=35$. Let's put on our algebra goggles and distribute that 8. We multiply 8 by $x$, which is $8x$, and then multiply 8 by $-7$, which gives us $-56$. So, after distributing, the equation becomes $8x - 56 = 35$. Now, the option suggests this is equivalent to $8x = 28$. To verify this, we need to see if adding 56 to both sides of $8x - 56 = 35$ results in $8x = 28$. Let's do the math: $8x - 56 + 56 = 35 + 56$. This simplifies to $8x = 91$. Comparing this to the stated equivalence, $8x = 28$, we see that $91$ is absolutely not equal to $28$. So, Option B is also a dud. It looks like the transformation here is incorrect as well. Perhaps a mistake was made when adding 35 and 56, or maybe the distribution itself was flawed. Let's double-check the distribution again: $8 * x = 8x$, and $8 * -7 = -56$. So, $8x - 56 = 35$. To get $8x$ by itself, we add 56 to both sides: $8x = 35 + 56$. Adding these numbers gives us $35 + 50 = 85$, and then add the remaining 6, $85 + 6 = 91$. So, $8x = 91$. The claim that it's equivalent to $8x = 28$ is false. This option fails the test. We're still on the hunt for that correct equation!

Analyzing Option C: $2(x-6)=-35$ is equivalent to $2x=-23$

Okay, team, let's get serious with Option C: $2(x-6)=-35$. Time to distribute the 2. We multiply 2 by $x$ to get $2x$, and then we multiply 2 by $-6$ to get $-12$. So, our distributed equation is $2x - 12 = -35$. The option claims this is equivalent to $2x = -23$. To check if this is true, we need to see if adding 12 to both sides of $2x - 12 = -35$ gives us $2x = -23$. Let's perform the addition: $2x - 12 + 12 = -35 + 12$. This simplifies to $2x = -23$. Now, let's compare this to the option's claim: $2x = -23$. Boom! They match perfectly! $ -35 + 12 $ is indeed $-23$. This means that Option C shows a correctly distributed and algebraically equivalent form of the original equation. We've found a potential winner, guys! This is exactly what we were looking for – a correct transformation of the equation. The distributive step $2(x-6)$ yields $2x - 12$, and then adding 12 to both sides of $2x - 12 = -35$ correctly results in $2x = -35 + 12$, which equals $2x = -23$. So, Option C is looking very good.

Analyzing Option D: $9(x-6)=-35$ is equivalent to $9x=-29$

Finally, let's examine Option D: $9(x-6)=-35$. We start by distributing the 9. So, $9 * x$ is $9x$, and $9 * -6$ is $-54$. Our distributed equation is $9x - 54 = -35$. The option states that this is equivalent to $9x = -29$. To verify, we need to add 54 to both sides of $9x - 54 = -35$. So, we have $9x - 54 + 54 = -35 + 54$. This simplifies to $9x = 19$. Let's compare this result, $9x = 19$, with the option's claim, $9x = -29$. Clearly, $19$ is not equal to $-29$. Therefore, Option D is also incorrect. It seems like a calculation error occurred when trying to find the new value on the right side of the equation after distribution. The distribution $9(x-6)$ correctly gives $9x-54$. To isolate $9x$, we add 54 to both sides: $9x = -35 + 54$. Calculating $-35 + 54$ gives us $54 - 35$. $54 - 30 = 24$, and $24 - 5 = 19$. So, $9x = 19$. The statement that it is equivalent to $9x = -29$ is false. Option D does not hold true.

The Verdict: Unveiling the Winning Ticket!

After meticulously breaking down each option, we can confidently declare the winning ticket. We performed the distributive property on each initial equation and then checked if the subsequent step shown in the option was a valid algebraic transformation to isolate the term with $x$. In Option A, we found $4x = 55$, not $4x = 40$. Option B led to $8x = 91$, not $8x = 28$. Option D resulted in $9x = 19$, not $9x = -29$. However, Option C, $2(x-6)=-35$, correctly transformed into $2x - 12 = -35$, and further correctly resulted in $2x = -23$ by adding 12 to both sides. Therefore, Option C is the winning ticket! It's the only statement that accurately represents an equivalent equation after the first step of simplification. Math problems like these are great for reinforcing fundamental algebraic skills. Keep practicing, keep questioning, and you'll master these concepts in no time. Don't forget to check back with Plastik Magazine for more math adventures and brain teasers. Stay curious, and happy solving!