Solve For (y² - 6y + 9) ÷ (3 - Y) When Y = 1 3/5

Hey math whizzes! Ever stared at an equation and thought, "What fresh algebra hell is this?" Well, gather 'round, because today we're diving deep into a problem that might look a little intimidating at first glance, but trust me, it's totally manageable. We're going to calculate the value of (y² - 6y + 9) ÷ (3 - y) when our variable y is equal to 1 3/5. This isn't just about crunching numbers, guys; it's about flexing those algebraic muscles and understanding how to simplify expressions before you even plug in the values. So, grab your calculators (or don't, we're going old school here!) and let's break this down.

Understanding the Expression: A Little Algebraic Magic

Before we even think about substituting y = 1 3/5, let's take a good, hard look at the expression we need to simplify: (y² - 6y + 9) ÷ (3 - y). Does that numerator, y² - 6y + 9, look familiar? If you've been around the algebra block a few times, you might recognize this as a perfect square trinomial. Remember the formula for $(a - b)²$? It's $a² - 2ab + b²$. In our case, if we let $a = y$ and $b = 3$, then $a² = y²$, $2ab = 2(y)(3) = 6y$, and $b² = 3² = 9$. Bingo! So, y² - 6y + 9 is actually just (y - 3)². This is a huge simplification, people! It means our original expression, (y² - 6y + 9) ÷ (3 - y), can be rewritten as (y - 3)² ÷ (3 - y). See how much cleaner that is already?

Now, let's look at the denominator, (3 - y). We also have (y - 3) in our simplified numerator. These two are almost the same, right? They're just negatives of each other. Remember that $(y - 3) = -(3 - y)$. So, we can rewrite (y - 3)² as (-(3 - y))². And when you square a negative, it becomes positive! So, (-(3 - y))² is the same as (3 - y)². Our expression now looks like (3 - y)² ÷ (3 - y). This is where the real magic happens, guys. As long as $(3 - y)$ is not equal to zero (which we'll check later), we can cancel out one of the (3 - y) terms from the numerator with the one in the denominator. This leaves us with just (3 - y). Ta-da! We've simplified the entire beast of an expression down to a single term: 3 - y. Isn't algebra awesome?

Plugging in the Value: Let's Get Numerical!

Okay, we've done the heavy lifting with the algebra, simplifying (y² - 6y + 9) ÷ (3 - y) down to (3 - y). Now comes the part where we actually use the given value of y = 1 3/5. First things first, let's convert that mixed number into an improper fraction. 1 3/5 is the same as $(1 * 5 + 3) / 5$, which equals 8/5. So, our value for y is 8/5.

Now, we just substitute this value into our simplified expression, 3 - y. So, we need to calculate 3 - 8/5. To do this, we need a common denominator. We can rewrite 3 as 15/5. So, the calculation becomes 15/5 - 8/5. Subtracting the numerators gives us (15 - 8) / 5, which equals 7/5.

Now, we need to check if our initial simplification was valid. We divided by $(3 - y)$, so we need to make sure $(3 - y)$ is not zero. If $y = 8/5$, then $3 - y = 3 - 8/5 = 15/5 - 8/5 = 7/5$. Since $7/5$ is not zero, our simplification was perfectly valid!

Finally, we need to express our answer 7/5 in the same format as the options provided. 7/5 as a mixed number is 1 2/5. And there you have it! The value of the expression is 1 2/5. It's always a good idea to simplify the algebraic expression first, because it makes the substitution and calculation so much easier, and often prevents silly arithmetic errors. Remember this trick, guys; it's a lifesaver in many math problems!

Checking the Options and Final Answer

So, we’ve done the math and arrived at our answer: 1 2/5. Now, let's take a peek at the options provided to make sure we're on the right track and that our answer matches one of them. The options are:

A) 1 2/5 B) -5 3/5 C) -2/5 D) 1 3/5 E) 5 3/5

Lookie here! Our calculated value, 1 2/5, directly matches option A. This gives us a big confidence boost that our simplification and substitution steps were spot on. It’s always super satisfying when your hard work lines up perfectly with one of the choices, right? This confirms that our algebraic manipulation, specifically recognizing the perfect square trinomial and the relationship between $(y - 3)$ and $(3 - y)$, paid off big time. Had we tried to plug in $y = 8/5$ directly into the original expression, the calculations would have been significantly more complex and prone to errors. Imagine squaring $8/5$, multiplying it by $6$, and then adding or subtracting these larger numbers – definitely more chances to mess up!

This problem really highlights the power of algebraic simplification. It's not just an academic exercise; it's a practical tool that makes complex problems much more approachable. By understanding the structure of the expression, we were able to reduce it to its simplest form, (3 - y), before introducing the numerical value. This strategy is a key skill in mathematics, applicable far beyond this single problem. So, the final, confirmed answer is A) 1 2/5. Great job working through this one, everyone!