G3.4 若 $a 、 b$ 及 $y$ 為實數, 並满足 $\left\{\begin{array}{l}a+b+y=5 \\ a b+b y+a y=3\end{array}\right.$, 求 $y$ 的最大值。
If $a, b$ and $y$ are real numbers and satisfy $\left\{\begin{array}{l}a+b+y=5 \\ a b+b y+a y=3\end{array}\right.$, determine the greatest possible value of $y$.
From (1): $a=5-(b+y)$
Sub. into (2): $[5-(b+y)] b+b y+[5-(b+y)] y=3$ $b^{2}+(y-5) b+\left(y^{2}-5 y+3\right)=0$, this is a quadratic equation in $b$.
For real values of $b, \Delta=(y-5)^{2}-4\left(y^{2}-5 y+3\right) \geq 0$
$$
\begin{array}{l}
-3 y^{2}+10 y+13 \geq 0 \\
3 y^{2}-10 y-13 \leq 0 \\
(3 y-13)(y+1) \leq 0 \\
-1 \leq y \leq \frac{13}{3}
\end{array}
$$

The maximum value of $y=\frac{13}{3}$.