To solve the rational equation
[tex]\[
\frac{p}{7} - \frac{p}{(p-9)} = \frac{9}{(9-p)},
\][/tex]
we follow these steps:
1. Identify the denominators: The denominators in the equation are [tex]\(7\)[/tex], [tex]\(p-9\)[/tex], and [tex]\(9-p\)[/tex].
2. Recognize the relationship between [tex]\(p-9\)[/tex] and [tex]\(9-p\)[/tex]:
[tex]\[
9 - p = -(p - 9).
\][/tex]
So, we can rewrite the right-hand side as:
[tex]\[
\frac{9}{(9-p)} = -\frac{9}{(p-9)}.
\][/tex]
Therefore, our equation becomes:
[tex]\[
\frac{p}{7} - \frac{p}{(p-9)} = -\frac{9}{(p-9)}.
\][/tex]
3. Combine the terms on the left-hand side by finding a common denominator:
The common denominator for [tex]\(7\)[/tex] and [tex]\(p-9\)[/tex] is [tex]\(7(p-9)\)[/tex]. Thus:
[tex]\[
\frac{p(p-9)}{7(p-9)} - \frac{7p}{7(p-9)} = -\frac{9 \times 7}{7(p-9)}.
\][/tex]
Simplify and combine terms:
[tex]\[
\frac{p(p-9) - 7p}{7(p-9)} = \frac{-63}{7(p-9)}.
\][/tex]
Combine the terms in the numerator:
[tex]\[
\frac{p^2 - 9p - 7p}{7(p-9)} = \frac{-63}{7(p-9)},
\][/tex]
which simplifies to:
[tex]\[
\frac{p^2 - 16p}{7(p-9)} = \frac{-63}{7(p-9)}.
\][/tex]
4. Eliminate the denominators by multiplying both sides of the equation by [tex]\(7(p-9)\)[/tex]:
[tex]\[
p^2 - 16p = -63.
\][/tex]
5. Form a quadratic equation:
[tex]\[
p^2 - 16p + 63 = 0.
\][/tex]
6. Solve the quadratic equation: To solve [tex]\(p^2 - 16p + 63 = 0\)[/tex], we can use the quadratic formula, [tex]\(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -16\)[/tex], and [tex]\(c = 63\)[/tex].
First, calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-16)^2 - 4 \cdot 1 \cdot 63 = 256 - 252 = 4.
\][/tex]
Then use the quadratic formula:
[tex]\[
p = \frac{16 \pm \sqrt{4}}{2 \cdot 1} = \frac{16 \pm 2}{2}.
\][/tex]
This results in two solutions:
[tex]\[
p = \frac{16 + 2}{2} = 9 \quad \text{and} \quad p = \frac{16 - 2}{2} = 7.
\][/tex]
7. Check the potential solutions for extraneous roots:
- For [tex]\(p = 9\)[/tex]:
Substitute [tex]\(p = 9\)[/tex] back into the original equation:
[tex]\[
\frac{9}{7} - \frac{9}{0} = \frac{9}{0},
\][/tex]
which is undefined due to division by zero. Hence, [tex]\(p = 9\)[/tex] is not a valid solution.
- For [tex]\(p = 7\)[/tex]:
Substitute [tex]\(p = 7\)[/tex] back into the original equation:
[tex]\[
\frac{7}{7} - \frac{7}{(7-9)} = \frac{9}{(9-7)},
\][/tex]
[tex]\[
1 + \frac{7}{2} = \frac{9}{2}.
\][/tex]
Simplifying both sides:
[tex]\[
1 + \frac{7}{-2} = -3.5.
\][/tex]
This correctly simplifies to [tex]\(4.5=4.5\)[/tex], making [tex]\(p = 7\)[/tex] a valid solution.
Thus, the solution set is [tex]\(\{7\}\)[/tex].
