Question - Solving an Equation with Exponential and Linear Terms
Solution:
The equation in the image is $$ e^{x} + 7x + 11 = 20 $$. To solve for $$ x $$, follow these steps:1. First, move constants to one side to isolate the $$ x $$-terms:\[ e^{x} + 7x + 11 - 20 = 0 \]\[ e^{x} + 7x - 9 = 0 \]2. This equation does not have a straightforward algebraic solution because it combines an exponential function and a linear term. To solve for $$ x $$, we typically use numerical methods such as the Newton-Raphson method or by graphically finding the intersection point.For a simple approximate solution, we can try a few guesses:- If $$ x = 0 $$, $$ e^0 + 7(0) - 9 = 1 + 0 - 9 = -8 $$, which is too low.- If $$ x = 1 $$, $$ e^1 + 7(1) - 9 = e + 7 - 9 $$. Given that $$ e \approx 2.718 $$, this is $$ 2.718 + 7 - 9 \approx 0.718 $$, which is too high.Since the function $$ e^{x} + 7x - 9 $$ is continuous, there will be a root between 0 and 1. To find the exact root, you would use a numerical solver or iterate with guesses refining towards the point where the function equals zero.
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