Solve. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

Key takeaway: This equation is a rational equation because the variable appears in a denominator. The key steps are to identify excluded values, clear denominators, and verify the result.

Step 1: Identify the restriction

The denominator is $a-9$, so

$a\ne 9$

Step 2: Multiply both sides by the LCD

The LCD is $a-9$. Multiply every term by $a-9$:

$\frac{-9}{a-9}(a-9)=\left(5-\frac{a}{a-9}\right)(a-9)$

This gives

$-9=5(a-9)-a$

Step 3: Simplify and solve

$-9=5a-45-a$ $-9=4a-45$ $36=4a$ $a=9$

Step 4: Check the restriction

But $a=9$ is not allowed because it makes the denominator zero. So it is an extraneous solution.

Answer: NO SOLUTION

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Pitfalls the pros know 👇 A common mistake is to stop at $a=9$ without checking the restriction $a\ne 9$. For rational equations, any value that makes a denominator zero must be rejected.

What if the problem changes? If the equation had a different denominator, the same process would apply: find the LCD, multiply through, solve the resulting linear equation, and then test each answer against the original denominators.

Tags: rational equation, LCD, extraneous solution