Multitasking with algebra

Algebra Level 2

$\begin{cases} \cfrac{y}{\sqrt{x}}=\sqrt{7}\\ e^x=\cfrac{y}{x}\\ \end{cases}$

Real numbers $x$ and $y$ , satisfying the system of equations above, can be written as $x = a W(b)$ and $y = a(b W(b))^a$ , where $a$ and $b$ are also real numbers. Find $a + b$ .

Notation: $W(\cdot)$ denotes the Lambert W-function .

The answer is 14.5.

1 solution

Chew-Seong Cheong
Jul 3, 2020

It is given that $\begin{cases} \dfrac y{\sqrt x} = \sqrt 7 & ...(1) \\ e^x = \dfrac yx \end{cases}$ . From $(1)$ , we have $y = \sqrt{7x}$ , substituting in $(2)$ ,

$\begin{aligned} e^x & = \sqrt{\frac 7x} \\ e^{2x} & = \frac 7x \\ xe^{2x} & = 7 \\ \text W \left(2xe^{2x} \right) & = \text W(14) \\ \implies 2x & = \text W(14) & \small \blue{\text{See reference}} \\ x & = \frac 12 \text W (14) \\ \implies y & = \sqrt{7x} = \frac 12 \left(14 \text W(14) \right)^\frac 12 \end{aligned}$

Therefore, $a+b = \frac 12 + 14 = \boxed{14.5}$ .

Reference: What is Lambert W-function?

Multitasking with algebra

The answer is 14.5.

1 solution

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