You can multiply both sides of an equation by a non-zero constant.
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Without an equality sign the given expression is not an equation
It means that whatever you have substituted is the solution of the given linear equation. Or you have substituted the equation in itself.
the Bratu's equation is a differential equation which is non-linear (such as, if we have some solutions for it, a linear combinaison of these solutions will not be everytime a solution). It's given by the equation y''+a*e^y=0 or d²y/dy² =-ae^y.
Without an equality sign the given expression can't be considered to be a straight line equation.
Without an equality sign and not knowing the plus or minus values of the given terms it cannot be considered to be a linear equation and so therefore there is no solution to it.