To solve the equation
\[ \sqrt{3x - 15} - 13 = -10, \]
we first isolate the square root term. We can do this by adding 13 to both sides:
\[ \sqrt{3x - 15} = -10 + 13. \]
This simplifies to:
\[ \sqrt{3x - 15} = 3. \]
Next, we square both sides to eliminate the square root:
\[ 3x - 15 = 3^2. \]
This simplifies to:
\[ 3x - 15 = 9. \]
Now, add 15 to both sides:
\[ 3x = 9 + 15. \]
This simplifies to:
\[ 3x = 24. \]
Finally, divide both sides by 3:
\[ x = \frac{24}{3} = 8. \]
Thus, the solution is
\[ \boxed{8}. \]
To verify, we can substitute \( x = 8 \) back into the original equation:
\[ \sqrt{3(8) - 15} - 13 = -10. \]
Calculating inside the square root:
\[ 3(8) - 15 = 24 - 15 = 9, \]
so we have:
\[ \sqrt{9} - 13 = -10. \]
This simplifies to:
\[ 3 - 13 = -10, \]
which is true. Therefore, the solution \( x = 8 \) is correct.
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