Given 𝑦 = 𝑎𝑥 − 2𝑎^2 and 𝑎 = 2−𝑥
3 , express y in terms of a

Given that \( y = ax - 2a^2 \) and \( a = \frac{2 - x}{3} \), we want to express \( y \) in terms of \( a \).

1. From the second equation, \( a = \frac{2 - x}{3} \), solve for \( x \):
   \[
   a = \frac{2 - x}{3} \implies 3a = 2 - x \implies x = 2 - 3a
   \]
2. Now substitute \( x = 2 - 3a \) into the expression for \( y \):
   \[
   y = a(2 - 3a) - 2a^2
   \]
3. Simplify the expression:
   \[
   y = a(2 - 3a) - 2a^2 = 2a - 3a^2 - 2a^2
   \]
   Combine like terms:
   \[
   y = 2a - 5a^2
   \]

Thus, \( y = 2a - 5a^2 \).

Would you like further details or clarifications on this?

Here are some related questions for practice:
1. How would you solve for \( x \) in terms of \( y \) and \( a \)?
2. What is the value of \( y \) when \( a = 1 \)?
3. How do you find the maximum or minimum value of \( y \) as a function of \( a \)?
4. Can you differentiate \( y = 2a - 5a^2 \) with respect to \( a \)?
5. How does changing \( a \) affect the slope of the line described by the equation \( y = 2a - 5a^2 \)?

**Tip:** Always simplify expressions by combining like terms to make equations more manageable.

Given 𝑦 = 𝑎𝑥 − 2𝑎^2 and 𝑎 = (2 − 𝑥)/3, express y in terms of a.

Learn how to express y in terms of a, given the equations y = ax - 2a^2 and a = (2 - x)/3. This problem involves algebraic substitution and simplification, perfect for students in Grades 9-12.

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