Unlock The Math: Equivalent Equations Explained

Hey guys! Today, we're diving deep into the awesome world of algebra to tackle a common challenge: finding equivalent equations. You know, those tricky versions of an equation that look totally different but have the exact same solution? It's like having different outfits for the same person – they might appear unique, but at their core, they're identical. We're going to break down a specific problem from Plastik Magazine, focusing on the equation rac{(5 x+6)}{2}=3-(4 x+12) and figuring out which of the given options are its true mathematical twins. So, grab your calculators, dust off your algebra skills, and let's get this done!

What Does 'Equivalent Equation' Really Mean?

Before we jump into solving, let's make sure we're all on the same page about what an equivalent equation is. In mathematics, two equations are considered equivalent if they have the same set of solutions. This means that whatever value of 'x' makes the original equation true will also make the equivalent equation true, and vice versa. Think of it as a secret handshake; only the right 'x' value knows it.

How do we get to these equivalent forms? We use a set of legal algebraic moves. These include:

• Adding or subtracting the same value to both sides of the equation.
• Multiplying or dividing both sides of the equation by the same non-zero value.
• Simplifying one or both sides of the equation without changing its value.

It's crucial to remember that whatever you do to one side of the equation, you must do to the other to maintain the balance. Messing this up is like trying to balance a see-saw by only pushing one side – it's just not going to work!

Breaking Down the Original Equation

Our main equation for today is rac{(5 x+6)}{2}=3-(4 x+12). This looks a bit intimidating, right? It's got fractions and parentheses, the usual suspects in algebra. To find its equivalents, we first need to simplify it as much as possible. Let's start by distributing the negative sign on the right side:

rac{(5 x+6)}{2} = 3 - 4x - 12

Now, let's combine the constant terms on the right side:

rac{(5 x+6)}{2} = -4x - 9

This is a simplified form of our original equation. We could also get rid of the fraction by multiplying both sides by 2:

2 * rac{(5 x+6)}{2} = 2 * (-4x - 9)

$(5x+6) = -8x - 18$

So, we've found two simplified forms that are definitely equivalent to our original equation: rac{(5 x+6)}{2} = -4x - 9 and $5x+6 = -8x - 18$. Keep these in your back pocket, as they'll be super handy when we look at the options!

Evaluating the Options: Are They Twins or Strangers?

Now, let's put on our detective hats and examine each option (A, B, C, D, E) to see if it's a true equivalent of our original equation. We'll use the simplified forms we found to make our comparisons easier. Remember, the goal is to see if they have the same solutions.

Option A: $5 x+6=3-(4 x+12)$

This looks like a step we took right at the beginning, before even simplifying. Let's see what happens if we simplify this one:

$5x+6 = 3 - 4x - 12$

$5x+6 = -4x - 9$

Is this equivalent to our original? Not quite. The original equation was rac{(5 x+6)}{2}=3-(4 x+12). Option A is missing the division by 2 on the left side. If we compare $5x+6 = -4x - 9$ with our original rac{(5 x+6)}{2}=3-(4 x+12), we can see they are different. If we were to multiply the original equation by 2, we'd get $5x+6 = 2(3 - (4x+12))$, which simplifies to $5x+6 = 6 - 8x - 24$, or $5x+6 = -8x - 18$. Option A, $5x+6 = -4x - 9$, does not match this. So, Option A is NOT equivalent. It's like a distant cousin, not a twin.

Option B: rac{(5 x+6)}{2}=-4 x-9

Guys, remember when we simplified the original equation by combining the terms on the right side? We got rac{(5 x+6)}{2} = -4x - 9. Bingo! This is exactly one of the simplified forms we found earlier. Since this equation was derived directly from the original using valid algebraic steps (distributing the negative and combining like terms), it must have the same solution set. Therefore, Option B IS equivalent. High five!

Option C: rac{5}{2} x+3=3-(4 x+12)

Let's simplify the left side of this equation. rac{5}{2}x + 3 is the same as rac{5x}{2} + rac{6}{2}. If we combine these terms over a common denominator, we get rac{5x+6}{2}. So, the left side of Option C, rac{5}{2} x+3, is indeed equivalent to rac{(5 x+6)}{2}. Now let's look at the right side: $3-(4x+12)$. This is also identical to the right side of our original equation. Since both sides of Option C are algebraically identical to the sides of our original equation, Option C IS equivalent. Another one in the bag!

Option D: rac{(5 x+6)}{2}=-4 x+15

Let's compare this to our simplified original equation, which was rac{(5 x+6)}{2} = -4x - 9. The left sides match perfectly. However, the right sides are different: $-4x+15$ versus $-4x-9$. The constant terms are not the same ($15 eq -9$). This difference means that the equation will yield a different solution for 'x'. Therefore, Option D is NOT equivalent. It's a imposter!

Option E: $5 x+6=-8 x-18$

Remember earlier, when we multiplied our original equation by 2 to get rid of the fraction? We ended up with $5x+6 = -8x - 18$. This is exactly Option E! Since multiplying both sides of an equation by a non-zero number (in this case, 2) is a valid operation that preserves the solution set, this equation is definitely equivalent. So, Option E IS equivalent. We found our third twin!

The Verdict: Our Trio of Equivalent Equations

After carefully analyzing each option, we've identified the equations that are true mathematical twins to our original equation rac{(5 x+6)}{2}=3-(4 x+12). These are the ones that, no matter how you rearrange them, will always give you the same answer for 'x'.

Drumroll, please...

• Option B: rac{(5 x+6)}{2}=-4 x-9 (This was our first simplified form)
• Option C: rac{5}{2} x+3=3-(4 x+12) (Both sides were shown to be identical to the original)
• Option E: $5 x+6=-8 x-18$ (This was our form after clearing the fraction)

So, the three equations equivalent to the given equation are B, C, and E. Great job working through this, everyone! Understanding equivalent equations is a fundamental skill in algebra that will make tackling more complex problems a breeze. Keep practicing, and you'll be an algebra whiz in no time!

Final Answer: The equations equivalent to rac{(5 x+6)}{2}=3-(4 x+12) are B, C, and E.