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I know you can plug in 1 and get the answer, but im curious as to why some websites with algebra calculators cannot solve the equation unless the website has some sort of graphic calculator also assigned to the problem?

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Polynomials and exponentials do not play nicely together, so a routine algebraic solution will not work. People are good at finding clever solutions, but calculators are not. People are biased toward toward integer solutions, so can see that $1$ is a solution (somewhat) easily. Calculators do not have the same bias, so $1$ is no different from $0.998$ to them.

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To answer this question, first we need to know what an "algebra calculator" is.

"Algebra Calculator is a calculator that gives step-by-step help on algebra problems."
-- Math Papa's homepage

Many CAS like SymPy can handle symbolic calculations. Typing solve('2*E**x-3*x+1'), you'll get \begin{equation*}\left [ \frac{1}{3} - \operatorname{LambertW}{\left (- \frac{2 e^{\frac{1}{3}}}{3} \right )}\right ]\end{equation*} from the system. The reason is that this equation is solvable with the function (using $p = 2, a = 3, b = -1$ in $p^{x}=ax+b$, which has the solution $$x=-{\cfrac {W\left(-{\dfrac {\ln p}{a}}\,p^{-{b/a}}\right)}{\ln p}}-{\frac {b}{a}}).$$

These CAS doesn't necessarily involve graphing, but they can solve symbolically some algebraic equations with polynomials and exponentials together. Therefore, one can believe that they can display step-by-step solutions to such problems since machines find solutions in a step-by-step manner.

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Most calculators with a "solve" function (including many HP calculators) can solve this whether or not they have graphics.

Just enter this function, optionally enter an initial guess, and press solve.

No graphics needed - just smarts.

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