quation: $$ \frac{5 s}{s^{2}-81}+\frac{4}{s-9}=\frac{1}{s+9} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{5s}{s^{2} - 81} + \frac{4}{s - 9} = \frac{1}{s + 9}
$$

let's follow these steps:

### Step 1: Factor the Denominator

Notice that $$ s^{2} - 81 $$ is a difference of squares and can be factored:

$$
s^{2} - 81 = (s - 9)(s + 9)
$$

So, the equation becomes:

$$
\frac{5s}{(s - 9)(s + 9)} + \frac{4}{s - 9} = \frac{1}{s + 9}
$$

### Step 2: Identify the Least Common Denominator (LCD)

The denominators in the equation are:
- $$ (s - 9)(s + 9) $$
- $$ s - 9 $$
- $$ s + 9 $$

The **Least Common Denominator (LCD)** is $$ (s - 9)(s + 9) $$.

### Step 3: Rewrite Each Term with the LCD

Convert each fraction so that they all have the LCD as their denominator:

$$
\frac{5s}{(s - 9)(s + 9)} + \frac{4 \cdot (s + 9)}{(s - 9)(s + 9)} = \frac{1 \cdot (s - 9)}{(s - 9)(s + 9)}
$$

This simplifies to:

$$
\frac{5s + 4(s + 9)}{(s - 9)(s + 9)} = \frac{s - 9}{(s - 9)(s + 9)}
$$

### Step 4: Eliminate the Denominator

Since the denominators are the same on both sides, we can equate the numerators:

$$
5s + 4(s + 9) = s - 9
$$

### Step 5: Simplify and Solve for $$ s $$

Expand and combine like terms:

$$
5s + 4s + 36 = s - 9
$$
$$
9s + 36 = s - 9
$$

Subtract $$ s $$ from both sides:

$$
8s + 36 = -9
$$

Subtract 36 from both sides:

$$
8s = -45
$$

Divide both sides by 8:

$$
s = -\frac{45}{8}
$$

### Step 6: Verify the Solution

Ensure that the solution does not make any denominator zero:

- $$ s - 9 = -\frac{45}{8} - 9 = -\frac{117}{8} \neq 0 $$
- $$ s + 9 = -\frac{45}{8} + 9 = \frac{27}{8} \neq 0 $$

Since neither denominator is zero, the solution is valid.

### Final Answer

$$
s = -\frac{45}{8}
$$
$$ s = -\frac{45}{8} $$

quation: \( \frac{5 s}{s^{2}-81}+\frac{4}{s-9}=\frac{1}{s+9} \)