Question

If \( x, y, z \) are real numbers such that \( x + y + z = 5 \) and \( xy + yz + zx = 3 \), what is the largest value that \( x \) can have?

• A. \( \frac{5}{3} \)
• B. \( \sqrt{19} \)
• C. \( \frac{13}{3} \)
• D. None of these

Hint:

When dealing with equations involving sums and products of variables, try expressing unknowns in terms of others to simplify.

Answer 1

The Correct Option is C

We are given the system of equations: 1. \( x + y + z = 5 \) 2. \( xy + yz + zx = 3 \) We can express \( y + z = 5 - x \). Substituting into the second equation: \[ xy + yz + zx = 3 \Rightarrow x(y + z) + yz = 3. \] Substituting \( y + z = 5 - x \) into this equation: \[ x(5 - x) + yz = 3 \Rightarrow 5x - x^2 + yz = 3. \] Now, we use the identity \( (y + z)^2 = y^2 + z^2 + 2yz \), so we know: \[ (5 - x)^2 = y^2 + z^2 + 2yz. \] We substitute \( yz \) from the earlier equation to find the largest value of \( x \). After solving, we get the value of \( x \) as \( \frac{13}{3} \).