Solve the equation and state the solution set. Remember to identify the values of the variable for which the expressions in each rational equation are undefined.
StartFraction 3 Over a minus 3 EndFraction plus StartFraction 7 Over a plus 3 EndFraction
equalsStartFraction negative 42 Over a squared minus 9 EndFraction

To solve the given rational equation and find the solution set, we first need to identify where the expressions are undefined and then solve the equation step-by-step.

### Step 1: Identify the Values of the Variable Where the Expressions are Undefined

The given equation is:

\[
\frac{3}{a - 3} + \frac{7}{a + 3} = \frac{-42}{a^2 - 9}
\]

1. **Undefined points** occur when the denominators are zero.
   
   - For \(\frac{3}{a - 3}\), the denominator \(a - 3 = 0 \implies a = 3\).
   - For \(\frac{7}{a + 3}\), the denominator \(a + 3 = 0 \implies a = -3\).
   - For \(\frac{-42}{a^2 - 9}\), the denominator \(a^2 - 9 = 0\).

2. **Factorize \(a^2 - 9\):**
   \[
   a^2 - 9 = (a - 3)(a + 3)
   \]

   The values \(a = 3\) and \(a = -3\) also make this expression undefined.

Therefore, the expressions are undefined when \(a = 3\) or \(a = -3\).

### Step 2: Solve the Equation

Rewrite the equation using the factorized form:

\[
\frac{3}{a - 3} + \frac{7}{a + 3} = \frac{-42}{(a - 3)(a + 3)}
\]

We want to combine the left-hand side over a common denominator:

\[
\frac{3(a + 3) + 7(a - 3)}{(a - 3)(a + 3)} = \frac{-42}{(a - 3)(a + 3)}
\]

Simplify the numerator on the left-hand side:

\[
3(a + 3) + 7(a - 3) = 3a + 9 + 7a - 21 = 10a - 12
\]

The equation becomes:

\[
\frac{10a - 12}{(a - 3)(a + 3)} = \frac{-42}{(a - 3)(a + 3)}
\]

Since the denominators are the same, we can equate the numerators:

\[
10a - 12 = -42
\]

### Step 3: Solve for \(a\)

Add 12 to both sides:

\[
10a = -42 + 12
\]

\[
10a = -30
\]

Divide by 10:

\[
a = -3
\]

### Step 4: Verify if \(a = -3\) is a Valid Solution

Recall that \(a = -3\) is a value where the expressions are undefined. Therefore, \(a = -3\) cannot be included in the solution set.

### Conclusion

There are no valid values of \(a\) that satisfy the given equation without making any of the expressions undefined. **The solution set is empty**.

### Summary

- The expressions are undefined for \(a = 3\) and \(a = -3\).
- Solving the equation gives \(a = -3\), but it is not valid since it makes the original expression undefined.
- **Solution Set:** \(\emptyset\).

Would you like more details on any step or have any other questions?

### Related Questions

1. How do you solve rational equations with different denominators?
2. What steps are necessary to factorize quadratic expressions?
3. What is the method to find the least common denominator for rational expressions?
4. How do you determine where a rational function is undefined?
5. What are asymptotes, and how do they relate to rational expressions?

**Tip:** Always check if the solution to a rational equation makes any denominator zero, as such values are excluded from the solution set.

Solve the equation and state the solution set. Remember to identify the values of the variable for which the expressions in each rational equation are undefined.
StartFraction 3 Over a minus 3 EndFraction plus StartFraction 7 Over a plus 3 EndFraction equals StartFraction negative 42 Over a squared minus 9 EndFraction.

Discover how to solve rational equations like 3/(a-3) + 7/(a+3) = -42/(a^2-9) while identifying undefined variable values. This detailed solution explores algebraic techniques, factoring, and rational equation principles. Ideal for Grades 9-11.

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