Mathematical Notes: The Solution and Resultant of Elimination

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\]This form of the solution conforms most closely to common sense: since \(x^{\prime}\) contains \(b\) and is contained in \(a^{\prime}\), it is natural that \(x\) should be equal to the sum of \(b\) and a part of \(a^{\prime}\) (that is to say, the part common to \(a^{\prime}\) and \(x\)). The solution is generally indeterminate (between the limits \(a^{\prime}\) and \(b\)); it is determinate only when the limits are equal,\[a^{\prime}=b,\]for then\[x=b+a^{\prime}x=b+bx=b=a^{\prime}.\]Then the equation assumes the form\[(ax+a^{\prime}x^{\prime}=0)=(a^{\prime}=x)\]and is equivalent to the double inclusion\[(a^{\prime} indeterminate _.We shall reach the same conclusion if we observe that (\(a+b\)) is the superior limit of the function \(ax+bx\) and that, if this limit is 0, the function is necessarily zero for all values of \(x\),\[(ax+bx^{\prime}


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