Linear left side - exponential right side

Algebra Level 3

One of the solutions of the equation 4 x + 7 = e x 4x+7=e^x can be expressed in the form − 7 / 4 − W ( a ) -7/4-W(a) , where W ( x ) W(x) is the inverse function of y = x e x y=xe^x for x ≥ − 1 x\geq -1 . Find the approximate value of a a rounded to three decimal places.


The answer is -0.043.

Arturo Presa by Arturo Presa , 62, USA

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1 solution

Arturo Presa
Mar 17, 2019

By replacing the solution − 7 4 − W ( a ) -\frac{7}{4} -W(a) into the equation, we get 4 ( − 7 4 − W ( a ) ) + 7 = e − 7 4 − W ( a ) . 4(-\frac{7}{4} -W(a))+7=e^{-\frac{7}{4}-W(a)}. Simplifying, we obtain, − 4 W ( a ) = e − 7 4 e − W ( a ) , -4W(a)=e^{-\frac{7}{4}}e^{-W(a)}, or, equivalently, a = W ( a ) e W ( a ) = − 1 4 e − 7 4 . a=W(a)e^{W(a)}= -\frac{1}{4} e^{-\frac{7}{4}}. Therefore, a ≈ − 0.043. a\approx\boxed{-0.043.}

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