Question

Infinitely many solutions
3x - 9 = x - 9

Answer 1

To solve the equation \(3x - 9 = x - 9\), we can begin by isolating \(x\).

1. Start with the original equation:
\[
3x - 9 = x - 9
\]

2. Subtract \(x\) from both sides:
\[
3x - x - 9 = -9
\]

3. This simplifies to:
\[
2x - 9 = -9
\]

4. Now, add \(9\) to both sides:
\[
2x - 9 + 9 = -9 + 9
\]

5. Simplifying gives:
\[
2x = 0
\]

6. Finally, divide both sides by \(2\):
\[
x = 0
\]

Now, checking if this solution makes sense in terms of the original equation:

- Substitute \(x = 0\) back into the original equation:
\[
3(0) - 9 = 0 - 9
\]
\[
-9 = -9
\]

Since both sides are equal, it confirms that \(x = 0\) is indeed a solution.

However, to explore the possibility of infinitely many solutions, we can rearrange the original equation and see if it simplifies to an identity.

Rearranging:
\[
3x - 9 = x - 9
\]
Subtracting both sides:
\[
3x - x - 9 + 9 = 0
\]
This simplifies to:
\[
2x = 0
\]
Which further simplifies to:
\[
x = 0
\]

This means the equation does not lead to infinitely many solutions (which would occur if we arrived at an identity, like \(0 = 0\)). Instead, it shows that there is exactly one solution, which is \(x = 0\).

Thus, the equation \(3x - 9 = x - 9\) has **one unique solution**:
\[
\boxed{0}
\]