Solve for x e^(1-3x)=3-2x

Problem

e^(1−3*x)=3−2*x

Solution

1. Identify the type of equation. This is a transcendental equation where the variable x appears both in the exponent and in a linear term, meaning it cannot be solved using standard algebraic methods.

2. Apply the Lambert W function, which is defined as the inverse of ƒ(w)=w*e^w To use it, we must manipulate the equation into the form ƒ(x)*e^ƒ(x)=C

3. Rearrange the equation by multiplying both sides by e^(3*x) to move the exponential term to the other side.

1=(3−2*x)*e^(3*x−1)

4. Multiply both sides by e^3.5 or adjust the linear term to match the exponent. Let u=3−2*x then x=(3−u)/2 Substituting this into the exponent 3*x−1 gives 3*((3−u)/2)−1=(9−3*u−2)/2=(7−3*u)/2

5. Observe that because the linear part and the exponent have different coefficients for x (−2 vs −3, the solution involves a complex argument for the Lambert W function or numerical approximation.

6. Use numerical methods such as Newton's Method to find the roots. Let ƒ(x)=e^(1−3*x)+2*x−3

7. Evaluate the function at x=1 ƒ(1)=e^(−2)+2−3≈0.135−1=−0.865

8. Evaluate the function at x=0.5 ƒ(0.5)=e^(−0.5)+1−3≈0.606−2=−1.394

9. Evaluate the function at x=1.5 ƒ(1.5)=e^(−3.5)+3−3≈0.03

10. Refine the root near x=1.5 using iteration. The value where the curve crosses zero is approximately 1.488

Final Answer

x≈1.488

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