https://file.aiquickdraw.com/mathbot/file/17261546824990mvz3wtp.jpg

The equation provided in the image is:

\[
-4 + \frac{1}{y-3} = \frac{4}{(y+1)(y-3)}
\]

Let's solve for \( y \).

### Step 1: Move \( -4 \) to the right-hand side:
\[
\frac{1}{y-3} = \frac{4}{(y+1)(y-3)} + 4
\]

### Step 2: Find a common denominator and combine terms on the right side:
To add \( 4 \), we need to express it as a fraction with the denominator \( (y+1)(y-3) \).

\[
4 = \frac{4(y+1)}{(y+1)(y-3)}
\]

Now, the equation becomes:
\[
\frac{1}{y-3} = \frac{4 + 4(y+1)}{(y+1)(y-3)}
\]

### Step 3: Simplify the right-hand side:
\[
\frac{1}{y-3} = \frac{4 + 4y + 4}{(y+1)(y-3)}
\]
\[
\frac{1}{y-3} = \frac{4y + 8}{(y+1)(y-3)}
\]

### Step 4: Cross-multiply:
\[
1 \cdot (y+1)(y-3) = (y-3)(4y + 8)
\]

Cancel out the common factor \( (y-3) \) on both sides, assuming \( y \neq 3 \):
\[
1 \cdot (y+1) = 4y + 8
\]

### Step 5: Simplify and solve for \( y \):
\[
y + 1 = 4y + 8
\]
\[
1 - 8 = 4y - y
\]
\[
-7 = 3y
\]
\[
y = -\frac{7}{3}
\]

### Solution:
The solution to the equation is \( y = -\frac{7}{3} \).

Would you like further details on this solution, or have any other questions? Here are five related problems:

1. Solve: \( \frac{2}{y-1} + 1 = \frac{3}{(y+2)(y-1)} \).
2. Solve for \( y \): \( \frac{1}{y+2} + 5 = \frac{3}{y^2-4} \).
3. How do you check for extraneous solutions in rational equations?
4. Solve: \( \frac{5}{y-4} = \frac{6}{y+2} \).
5. What are the steps to solve a rational equation involving multiple fractions?

**Tip**: Always check the excluded values of \( y \) by examining the denominators in rational equations!

Solve the equation: -4 + 1/(y-3) = 4/((y+1)(y-3))

Explore how to solve the rational equation -4 + 1/(y-3) = 4/((y+1)(y-3)) by isolating terms, cross-multiplying, and simplifying. Understand key algebraic concepts like rational equations and factoring, suitable for high school students in Grades 9-12.

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