LaTeX practice

Question

LaTeX practice

Aratox · Answer

To solve the equation $\frac{1}{n-8} - 1 = \frac{7}{n-8}$, we will follow a systematic approach, including checking for extraneous solutions.

### Step 1: Combine the terms on the left side

We first want to get a common denominator on the left side. The left side can be rewritten as:
$$
\frac{1}{n-8} - 1 = \frac{1}{n-8} - \frac{n-8}{n-8} = \frac{1 - (n-8)}{n-8} = \frac{1 - n + 8}{n-8} = \frac{9 - n}{n-8}
$$

So, the equation now looks like:
$$
\frac{9 - n}{n-8} = \frac{7}{n-8}
$$

### Step 2: Eliminate the denominators

Since the denominators are the same, we can multiply both sides by $n - 8$ (assuming $n \neq 8$):
$$
9 - n = 7
$$

### Step 3: Solve for $n$

Now we can solve for $n$:
$$
9 - n = 7
$$
Subtract $9$ from both sides:
$$
-n = 7 - 9
$$
$$
-n = -2
$$
Multiply both sides by $-1$:
$$
n = 2
$$

### Step 4: Check for extraneous solutions

We need to check if our solution $n = 2$ satisfies the original equation. We substitute $n = 2$ back into the original equation:

The original equation is:
$$
\frac{1}{n-8} - 1 = \frac{7}{n-8}
$$

Substituting $n = 2$:
$$
\frac{1}{2-8} - 1 = \frac{7}{2-8}
$$
$$
\frac{1}{-6} - 1 = \frac{7}{-6}
$$
Convert $1$ into a fraction:
$$
\frac{1}{-6} - \frac{6}{6} = \frac{7}{-6}
$$
Now simplify the left side:
$$
\frac{1 - 6}{-6} = \frac{-5}{-6} = \frac{5}{6}
$$
The right side:
$$
\frac{7}{-6} = \frac{-7}{6}
$$

Since $\frac{5}{6} \neq \frac{-7}{6}$, $n = 2$ does not satisfy the original equation.

### Conclusion

The solution $n = 2$ is an extraneous solution. Since our procedure yielded no other possible values, we conclude that **there are no solutions** for the equation $\frac{1}{n-8} - 1 = \frac{7}{n-8}$.