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To solve part (a), we need to express \( y \) in terms of \( x \) from the equation:

\[
x + y = 230
\]

**Step for (a):**

1. Subtract \( x \) from both sides to isolate \( y \):
   \[
   y = 230 - x
   \]

Thus, the answer for part (a) is:

\[
y = 230 - x
\]

Would you like further assistance with the next steps or a deeper explanation?

Here are 5 related questions that could expand your understanding of optimization:

1. What method can we use to maximize the product \( P = xy \) in this scenario?
2. How do we apply derivatives to find the maximum value of a function?
3. What conditions need to be met to ensure \( x \) and \( y \) remain nonnegative?
4. How does rewriting one variable in terms of another help in solving optimization problems?
5. What are some other applications of maximizing a product given a sum constraint?

**Tip:** In optimization problems, always rewrite one variable in terms of the other when possible. This simplifies calculations and helps find optimal solutions effectively.

Follow the steps below to find the nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression. Complete parts (a) through (f) below.

x + y = 230 and the product P = xy as large as possible.

(a) Solve x + y = 230 for y.

Learn how to solve an optimization problem involving the sum and product of two nonnegative numbers. Find values of x and y that maximize the product P = xy, given x + y = 230. This problem covers algebraic manipulation, substitution, and optimization using derivatives. Suitable for high school students in Grades 9-12.

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