To solve for $y$ in the equation:

$\sqrt{y - 8} + 10 = 2$

we will isolate the square root term and then square both sides to eliminate the square root.

Step-by-Step Solution:

1. Isolate the Square Root Term: $\sqrt{y - 8} + 10 = 2$ Subtract 10 from both sides: $\sqrt{y - 8} = 2 - 10$ $\sqrt{y - 8} = -8$

2. Analyze the Result: The equation $\sqrt{y - 8} = -8$ has no solution, because the square root of a real number (which is always non-negative) cannot equal a negative number like $-8$.

Conclusion:

There is no solution for $y$ in the real numbers.

Would you like further details or have any questions?

Related Questions:

1. How would the solution change if the equation were $\sqrt{y - 8} + 10 = 12$?
2. What are the properties of square root functions that restrict them from equaling negative numbers?
3. How would this problem change if we allowed complex numbers as solutions?
4. What other transformations could be used to solve equations involving square roots?
5. How do we handle equations with multiple square roots on both sides?

Tip:

Always check the domain of square root terms to ensure that the equation is valid for real solutions.