Simplifying Expressions (3y-4)(2y+7)+11y-9 Finding Equivalency

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Hey there, math enthusiasts! Ever stumbled upon an expression that looks like a tangled mess? Well, today we're diving headfirst into one of those – the intriguing $(3y-4)(2y+7)+11y-9$. Our mission? To simplify it and discover its true identity, or in math speak, its equivalent expression. So, buckle up, and let's embark on this mathematical adventure together!

The Challenge: Deciphering the Expression

Our journey begins with the given expression: $(3y-4)(2y+7)+11y-9$. At first glance, it might seem daunting, a jumble of parentheses, variables, and numbers. But don't worry, we're going to break it down step by step, just like a detective solving a case. The key here is to remember the order of operations (PEMDAS/BODMAS) and the distributive property. Think of it as our mathematical toolkit for this quest.

Step 1: Unleashing the Power of Distribution

The first part of our expression, $(3y-4)(2y+7)$, is a classic example where the distributive property shines. Remember, this property is like a mathematical handshake – each term in the first set of parentheses needs to 'shake hands' (multiply) with each term in the second set. So, let's get those hands shaking!

  • 3y multiplies with 2y: $3y * 2y = 6y^2$ (Don't forget, when multiplying variables with exponents, we add the exponents)
  • 3y multiplies with 7: $3y * 7 = 21y$
  • -4 multiplies with 2y: $-4 * 2y = -8y$
  • -4 multiplies with 7: $-4 * 7 = -28$

Now, let's put these pieces together. The expanded form of $(3y-4)(2y+7)$ is $6y^2 + 21y - 8y - 28$. We've successfully navigated the first hurdle!

Step 2: Taming the Like Terms

Our expression is starting to take shape, but it's not quite in its simplest form yet. Notice how we have a couple of terms with the same variable (y) – $21y$ and $-8y$? These are what we call 'like terms,' and they're begging to be combined. Think of it as grouping similar objects together to make them easier to count.

Combining $21y$ and $-8y$ is like a simple addition problem: $21y - 8y = 13y$. So, our expression now looks like this: $6y^2 + 13y - 28$. Much cleaner, right?

Step 3: Bringing in the Reinforcements

We're not done yet! Remember the original expression? We still have the $+11y-9$ part waiting in the wings. Let's bring it into the mix: $6y^2 + 13y - 28 + 11y - 9$.

Guess what? We have more like terms to combine! This time, it's the $13y$ and $11y$, and the constants $-28$ and $-9$. Let's get to it:

  • Combining $13y$ and $11y$: $13y + 11y = 24y$
  • Combining $-28$ and $-9$: $-28 - 9 = -37$

Step 4: The Grand Finale – Unveiling the Equivalent Expression

We've reached the final stage of our mathematical journey! After all the distribution and combining of like terms, we're left with: $6y^2 + 24y - 37$.

And there you have it, guys! We've successfully transformed the original expression into its equivalent form. It's like revealing the hidden beauty beneath a complex facade.

The Options: Spotting the Correct Match

Now that we've found our simplified expression, let's take a look at the options provided and see which one matches our masterpiece:

A. $9y-37$ B. $16y-6$ C. $6y^2+24y-37$ D. $6y^2+11y+19$

Drumroll, please…

The correct answer is **C. **$6y^2+24y-37$! It's the spitting image of our simplified expression. We did it!

Why Other Options Don't Make the Cut

It's always good to understand why the other options are incorrect. It reinforces our understanding of the process. So, let's briefly examine why options A, B, and D didn't make the grade:

  • **A. **$9y-37$: This option is missing the $y^2$ term, indicating an incomplete simplification.
  • **B. **$16y-6$: Similar to option A, this one also lacks the crucial $y^2$ term and doesn't accurately reflect the result of expanding and simplifying the original expression.
  • **D. **$6y^2+11y+19$: While this option has the $y^2$ term, the coefficients of the $y$ term and the constant term are incorrect, showing a mistake in the combining of like terms.

Key Takeaways: Mastering the Art of Simplification

So, what have we learned on this mathematical expedition? Here are some key takeaways to remember when tackling similar expressions:

  • The Distributive Property is Your Friend: Don't be intimidated by parentheses! The distributive property is your weapon of choice for expanding expressions.
  • Like Terms Unite: Always combine like terms to simplify your expression. It's like decluttering your math!
  • Order of Operations is the Law: Remember PEMDAS/BODMAS to ensure you're tackling the expression in the correct sequence.
  • Double-Check Your Work: Math is precise. Always take a moment to review your steps and make sure you haven't made any sneaky errors.

Practice Makes Perfect: Sharpening Your Skills

Like any skill, simplifying expressions becomes easier with practice. Try tackling similar problems on your own. You can even create your own expressions and challenge yourself to simplify them. The more you practice, the more confident you'll become in your mathematical abilities.

And remember, guys, math isn't about memorizing formulas; it's about understanding the process and developing problem-solving skills. So, embrace the challenge, have fun, and keep exploring the amazing world of mathematics!

In Conclusion: Triumph Over Complexity

We've successfully navigated the complexities of the expression $(3y-4)(2y+7)+11y-9$, simplified it, and identified its equivalent form. We've seen how the distributive property, combining like terms, and the order of operations are powerful tools in our mathematical arsenal. So, the next time you encounter a seemingly complicated expression, remember this journey, and know that you have the skills to unravel the mystery!