Certainly! Let's analyze and solve the given equation step by step:
The equation given is:
[tex]\[
\frac{1}{x - 3y} - \frac{y}{x^2 - 9y^2} = \frac{x + 2y}{x^2 - 9y^2}
\][/tex]
### Step 1: Identify and Simplify the Common Denominator
First, observe that the terms on both sides of the equation involve the expression [tex]\( x^2 - 9y^2 \)[/tex]. We can recognize that [tex]\( x^2 - 9y^2 \)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - 9y^2 = (x - 3y)(x + 3y)
\][/tex]
### Step 2: Express All Terms with the Common Denominator
We will now rewrite each term with the common denominator [tex]\( (x - 3y)(x + 3y) \)[/tex].
The left-hand side (LHS) of the equation is:
[tex]\[
\frac{1}{x - 3y} - \frac{y}{x^2 - 9y^2}
\][/tex]
Rewrite [tex]\(\frac{1}{x - 3y}\)[/tex] with the common denominator:
[tex]\[
\frac{1}{x - 3y} = \frac{1 \cdot (x + 3y)}{(x - 3y)(x + 3y)} = \frac{x + 3y}{(x - 3y)(x + 3y)}
\][/tex]
Now include the second term of LHS:
[tex]\[
-\frac{y}{x^2 - 9y^2} = -\frac{y}{(x - 3y)(x + 3y)}
\][/tex]
Combining these, LHS becomes:
[tex]\[
\frac{x + 3y}{(x - 3y)(x + 3y)} - \frac{y}{(x - 3y)(x + 3y)} = \frac{(x + 3y) - y}{(x - 3y)(x + 3y)}
\][/tex]
Simplify the numerator:
[tex]\[
(x + 3y) - y = x + 2y
\][/tex]
Therefore, the LHS simplifies to:
[tex]\[
\frac{x + 2y}{(x - 3y)(x + 3y)}
\][/tex]
### Step 3: Simplify the Right-Hand Side
The right-hand side (RHS) of the equation is already in a simplified form:
[tex]\[
\frac{x + 2y}{x^2 - 9y^2}
\][/tex]
Using our earlier factorization:
[tex]\[
\frac{x + 2y}{(x - 3y)(x + 3y)}
\][/tex]
### Step 4: Compare Both Sides
We now compare the simplified forms of both sides:
LHS:
[tex]\[
\frac{x + 2y}{(x - 3y)(x + 3y)}
\][/tex]
RHS:
[tex]\[
\frac{x + 2y}{(x - 3y)(x + 3y)}
\][/tex]
Since both sides are equal, the equation holds true for all [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that do not make the denominators zero (i.e., [tex]\( x \neq \pm 3y \)[/tex]).
### Final Conclusion
Hence, the left-hand side and the right-hand side of the equation are equal, confirming the equality:
[tex]\[
\frac{1}{x - 3y} - \frac{y}{x^2 - 9y^2} = \frac{x + 2y}{x^2 - 9y^2}
\][/tex]
This gives us our simplified and balanced result.
