If 𝑥=2x=2 and 𝑦=3y=3, then for what value of 𝑡t will the convex combination be 2.752.75?

A convex combination of \( x \) and \( y \) is given by:

\[
t \cdot x + (1 - t) \cdot y
\]

We are given that \( x = 2 \), \( y = 3 \), and we want to find the value of \( t \) such that the convex combination equals 2.75.

So, we have:

\[
t \cdot 2 + (1 - t) \cdot 3 = 2.75
\]

Now, let's solve for \( t \).

\[
2t + 3 - 3t = 2.75
\]

\[
- t + 3 = 2.75
\]

\[
- t = 2.75 - 3
\]

\[
- t = -0.25
\]

\[
t = 0.25
\]

Thus, the value of \( t \) that makes the convex combination equal to 2.75 is \( t = 0.25 \).

Would you like further details or have any questions? Here are some related questions you might explore:

1. What is a convex combination in linear algebra?
2. How can convex combinations be applied in optimization problems?
3. What are the geometric interpretations of convex combinations?
4. How do you derive the formula for a convex combination?
5. How would the solution change if the values of \( x \) and \( y \) were different?

**Tip:** Convex combinations are often used in interpolation between points, which is a key concept in computer graphics and data fitting.

Discover how to find the convex combination using the values x = 2, y = 3, and t = 0.25. Understand the concept of convex combinations in linear algebra with a step-by-step solution.

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