Question

Consider the equation 9/x−3=x2/x−3
.

Rewrite the equation as two equations joined by “and”. Explain the significance of the second equation.
Solve the equation and interpret your solution set.

Answer 1

We start with the given equation:

\[
\frac{9}{x-3} = \frac{x^2}{x-3}
\]

Since both sides of the equation have the common denominator \(x - 3\), we can rewrite the equation as:

\[
9 = x^2
\]

To rewrite this as two separate equations joined by "and," we note that the common denominator informs us of a restriction on the variable \(x\):

1. \(9 = x^2\) (the equation derived from the original equation)
2. \(x - 3 \neq 0\) (we must ensure the denominator is not zero)

The second equation (\(x - 3 \neq 0\)) is significant because it tells us that \(x\) cannot equal 3, to avoid division by zero in the original equation.

Next, we solve the first equation:

\[
9 = x^2
\]

Taking the square root of both sides gives us:

\[
x = 3 \quad \text{or} \quad x = -3
\]

Now we must consider the restriction from the second equation. The restriction \(x - 3 \neq 0\) tells us that \(x\) cannot be 3. Therefore, we discard the solution \(x = 3\).

The only solution we have that satisfies the equation without violating the restriction is:

\[
x = -3
\]

### Interpretation of the Solution Set

The solution set is:

\[
\{ -3 \}
\]

This means that the only value of \(x\) that satisfies the original equation while also respecting the restriction (that \(x\) cannot be 3) is \(x = -3\). Thus, -3 is the only valid solution to the equation.