Recall that the convex combination of two points in one dimension is given by:

𝑧=𝑡𝑥+(1−𝑡)𝑦𝑤ℎ𝑒𝑟𝑒 𝑡∈[0,1]


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

The convex combination of two points \( x \) and \( y \) in one dimension is given by:

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

where \( t \) is a parameter that lies in the interval \([0, 1]\).

Given \( x = 2 \), \( y = 3 \), and \( z = 2.75 \), we need to find the value of \( t \) such that:

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

Let's solve for \( t \):

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

\[
2.75 = -t + 3
\]

\[
-t = 2.75 - 3
\]

\[
-t = -0.25
\]

\[
t = 0.25
\]

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 happens if \( t = 0 \) or \( t = 1 \) in a convex combination?
2. How does the value of \( t \) affect the position of \( z \) between \( x \) and \( y \)?
3. Can the convex combination be used to represent points in higher dimensions?
4. What is the significance of the interval \( [0, 1] \) for \( t \) in a convex combination?
5. How would the problem change if \( x \) and \( y \) were negative?

**Tip:** In a convex combination, \( t \) represents the "weight" given to each point, and varying \( t \) moves the result along the line segment between the points \( x \) and \( y \).

Understand how to solve the convex combination problem using x = 2, y = 3, and find the value of t that yields z = 2.75. This problem explores concepts in linear algebra, including convex combinations and parametric equations.

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