Spot Bianca's Math Mistake: Equivalent Expressions

Hey guys, welcome back to Plastik Magazine! Today, we're diving into the wild world of math, specifically tackling algebraic expressions. We've got a little puzzle for you involving Bianca, who's trying to simplify an expression and, well, hit a bit of a snag. Let's break down her work step-by-step and figure out exactly where she went off track. Our main keyword here is Bianca's expression error, and we'll be exploring how to find equivalent expressions. So, grab your calculators (or just your brains!), and let's get this math party started!

The Expression and Bianca's First Move

Bianca's mission, should she choose to accept it, is to write an expression that is equivalent to $12+15(3-y)-10 y$. An equivalent expression is basically a different way of writing the same mathematical value. Think of it like having a different outfit on but still being the same person – the underlying value remains constant. Bianca starts with Step 1, which is simply the original expression: $12+15(3-y)-10 y$. So far, so good, right? No errors here, this is just the starting point. The challenge lies in how she manipulates this expression to simplify it. Remember, the goal is to end up with an expression that, no matter what number you plug in for 'y', will give you the same result as the original expression. This involves using the distributive property and combining like terms, concepts we'll see in action (or, in Bianca's case, mis-action) shortly. The expression itself has a constant term (12), a multiplication involving a binomial ($15(3-y)$), and another term with a variable ($-10y$). Each part needs careful handling to maintain equivalence. The distributive property is key here: multiplying the 15 by both terms inside the parentheses. Let's keep our eyes peeled for how she applies this crucial rule.

Step 2: The Point of Divergence

Now, let's move on to Step 2: $12+45+15 y-10 y$. This is where things get interesting, and potentially, a little bit messy. To get from Step 1 to Step 2, Bianca seems to have applied the distributive property to the $15(3-y)$ part. The distributive property states that $a(b+c) = ab + ac$. In Bianca's case, it looks like she's done $15 imes 3$ and $15 imes (-y)$. So, $15 imes 3$ should indeed be 45. That part is correct. However, $15 imes (-y)$ should result in $-15y$, not $+15y$. This is the crucial error, guys! She's incorrectly distributed the 15 across the subtraction, changing the sign of the 'y' term. Instead of $15 imes (-y) = -15y$, she's written $+15y$. This single sign change throws the entire expression off balance, making the new expression no longer equivalent to the original. This is a super common mistake, especially when dealing with negative signs. It's vital to remember that when you distribute a positive number (like 15) to a term that is being subtracted (like $-y$), the result should also be negative. Keep that in mind as we continue!

Step 3: The Final (Incorrect) Result

Finally, we arrive at Step 3: $57+5 y$. This step represents Bianca's attempt to combine the terms from Step 2. Let's see if she made any further errors or if this result stems solely from the mistake in Step 2. In Step 2, we have $12+45+15 y-10 y$. If we ignore the sign error for a moment and just focus on combining terms as Bianca did, she added the constants: $12 + 45 = 57$. This part is correct, assuming Step 2 was correct. Then, she combined the 'y' terms: $15y - 10y = 5y$. Again, if Step 2 were correct, this would also be correct. So, the calculation in Step 3 itself, based on the numbers presented in Step 2, is mathematically sound. However, because Step 2 contained a fundamental error in applying the distributive property (specifically with the sign of the 'y' term), Step 3, and thus the final simplified expression, is not equivalent to the original expression. The root of all the problems lies squarely in Step 2. This highlights how one small slip-up early on can cascade into a completely incorrect final answer. It’s a good reminder for all of us to double-check our work, especially during distribution and sign changes.

Identifying Bianca's Error

So, to recap and directly answer the question: Choose the step that shows her error? The error occurs in Step 2. This is where Bianca incorrectly applied the distributive property, changing the sign of the term involving 'y' from $-15y$ to $+15y$. The expression should have been $12 + 45 - 15y - 10y$. If she had done that correctly, the subsequent steps would lead to a different, and correct, equivalent expression. Let's quickly show what the correct simplification would look like:

Original Expression: $12+15(3-y)-10 y$

Step 1 (Correct): $12+15(3-y)-10 y$

Step 2 (Corrected): Apply distributive property: $15 imes 3 = 45$ and $15 imes (-y) = -15y$. So, the expression becomes $12 + 45 - 15y - 10y$.

Step 3 (Corrected): Combine like terms. Combine the constants: $12 + 45 = 57$. Combine the 'y' terms: $-15y - 10y = -25y$.

Correct Equivalent Expression: $57 - 25y$

See the difference? A single sign error in Step 2 led Bianca to a completely different final answer. It's a classic example of how important attention to detail is in mathematics. Always, always, always double-check those signs when distributing!