Simplifying Expressions: Finding The Domain And Avoiding Pitfalls

Hey there, math enthusiasts! Let's dive into a problem that combines algebraic simplification with a crucial concept: finding the domain of an expression. We'll be working with the expression $\frac{9 x^2+4 y^2-12 x y}{4 y^2-9 x^2}$. Our goal? To simplify this expression and then determine the domain of the simplified form. This might seem like a straightforward task, but it's packed with nuances that can trip you up if you're not careful. So, grab your pencils (or your favorite digital stylus), and let's get started!

Unveiling the Initial Expression and Our Strategy

Let's start by taking a good look at the expression $\frac{9 x^2+4 y^2-12 x y}{4 y^2-9 x^2}$. At first glance, it might seem a bit intimidating, but don't worry! Our strategy is simple: first, we'll try to simplify the expression. Then, we will find the domain of the simplified expression. This involves factoring, canceling common terms, and carefully considering any restrictions that arise from the denominator. Remember, the domain of an expression is the set of all possible values for the variables (in this case, x and y) for which the expression is defined. What does "defined" mean? It means the expression gives us a real, meaningful result. We need to be especially mindful of two things that can make an expression undefined: division by zero and, in some cases, taking the square root of a negative number (though that's not relevant here). The core idea is to find the values of x and y that won't cause trouble.

Step-by-Step Simplification: A Journey Through Algebra

Now, let's roll up our sleeves and simplify the expression $\frac{9 x^2+4 y^2-12 x y}{4 y^2-9 x^2}$. This involves a bit of algebraic manipulation, specifically factoring. Our aim is to break down both the numerator and the denominator into simpler components to see if we can cancel out any common factors. Factoring is a fundamental skill in algebra, and it's something that gets easier with practice. It's like a puzzle where we try to find the pieces that fit together perfectly. The more you work with factoring, the quicker you'll recognize patterns and the more confident you'll become.

Factoring the Numerator: A Perfect Square Trinomial

Let's start with the numerator: $9x^2 + 4y^2 - 12xy$. This expression looks like it might be factorable. If we rearrange the terms slightly, we get $9x^2 - 12xy + 4y^2$. Notice anything? This expression is a perfect square trinomial. We can rewrite it as $(3x - 2y)^2$. The key to recognizing a perfect square trinomial is identifying two terms that are perfect squares ($9x^2$ and $4y^2$) and a middle term that is twice the product of their square roots ($-12xy = 2 * (3x) * (-2y)$). Practice recognizing these patterns; they will save you a lot of time and effort.

Factoring the Denominator: Difference of Squares

Next, let's look at the denominator: $4y^2 - 9x^2$. This is a classic example of the difference of squares. We can rewrite it as $(2y)^2 - (3x)^2$. The difference of squares can always be factored into the product of the sum and difference of the square roots of the terms. So, we can factor the denominator as $(2y - 3x)(2y + 3x)$. The difference of squares is another pattern that you'll encounter frequently, so make sure you're comfortable with it. You'll often find yourself using it in various algebraic problems.

Putting It All Together: Simplifying the Expression

Now we have $\frac{(3x - 2y)^2}{(2y - 3x)(2y + 3x)}$. Observe that $(3x - 2y)$ and $(2y - 3x)$ are very similar, but they have opposite signs. We can rewrite $(3x - 2y)$ as $-(2y - 3x)$. Substituting that into our expression, we have $\frac{(-1 * (2y - 3x))^2}{(2y - 3x)(2y + 3x)} = \frac{(2y - 3x)^2}{(2y - 3x)(2y + 3x)}$. Now we can cancel the common factor of $(2y - 3x)$, assuming it's not equal to zero. This leaves us with $\frac{2y - 3x}{2y + 3x}$. This is our simplified expression.

Determining the Domain of the Simplified Expression

Okay, we've simplified the expression to $\frac{2y - 3x}{2y + 3x}$. Now, it's time to find its domain. Remember, the domain is the set of all values for x and y for which the expression is defined. The most common place where an expression can become undefined is when we divide by zero. So, we need to find the values of x and y that make the denominator equal to zero. It's all about avoiding the pitfalls that can make our expression blow up or become nonsensical.

Identifying Restrictions: Avoiding Division by Zero

In our simplified expression, the denominator is $2y + 3x$. We need to make sure that $2y + 3x \ne 0$. Let's solve this inequality for y: $2y \ne -3x$, which means $y \ne -\frac{3}{2}x$. This tells us that any pair of values (x, y) that satisfies this inequality is allowed. Any point on the line $y = -\frac{3}{2}x$ is excluded from the domain because it makes the denominator zero.

Considering the Original Expression: A Crucial Step

Now, let's go back to the original expression: $\frac{9 x^2+4 y^2-12 x y}{4 y^2-9 x^2}$. When we simplified, we canceled the term $(2y - 3x)$. This means that we implicitly assumed that $2y - 3x \ne 0$. So, we must add this restriction to our domain. Thus, $2y - 3x \ne 0$, or $y \ne \frac{3}{2}x$. This tells us that any pair of values (x, y) that satisfies this inequality is allowed. Therefore, we should exclude the line $y = \frac{3}{2}x$ from the domain.

Defining the Domain: The Final Answer

Therefore, the domain of the simplified expression $\frac{2y - 3x}{2y + 3x}$ is all real numbers (x, y) such that $y \ne -\frac{3}{2}x$ and $y \ne \frac{3}{2}x$. In set notation, this can be written as: {(x, y) | x, y ∈ ℝ, y ≠ -\frac{3}{2}x, y ≠ \frac{3}{2}x}. That's it! We've successfully simplified the expression, identified the restrictions, and determined the domain.

Conclusion: Mastering Domains and Simplification

So, there you have it! We've worked our way through simplifying the expression $\frac{9 x^2+4 y^2-12 x y}{4 y^2-9 x^2}$ and, more importantly, finding its domain. Remember that understanding domains is crucial when working with algebraic expressions. Always be mindful of potential restrictions caused by denominators, square roots, and any other operations that might lead to undefined results. Through this problem, we've strengthened our skills in factoring, canceling, and recognizing key algebraic patterns. The more you practice these concepts, the more confident and proficient you'll become in tackling mathematical challenges.

This journey has reinforced the significance of the following:

• Factoring: Breaking down expressions into manageable parts is crucial for simplification.
• Domain Restrictions: Identifying and accounting for values that make the denominator zero is essential.
• Attention to Detail: Carefully considering each step and the original expression helps avoid mistakes.

Keep practicing, keep exploring, and keep the mathematical spirit alive! You've got this!

For further learning, I would recommend checking out some reliable resources such as:

• Khan Academy: Khan Academy's Algebra Section This platform offers excellent courses, practice exercises, and video tutorials on algebra concepts.
• Purplemath: Purplemath's Algebra Lessons Provides comprehensive and well-explained algebra lessons.

Good luck! Keep up the great work, and happy learning! Remember, the world of mathematics is vast and exciting. There's always something new to discover.