Elimination of Unknown Quantities in Equations

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0.37 Elimination of Several Unknown QuantitiesWe shall now consider an equation involving several unknown quantities and suppose it reduced to the normal form, _i.e._, its first member developed with respect to the unknown quantities, and its second member zero. Let us first concern ourselves with the problem of elimination. We can eliminate the unknown quantities either one by one or all at once.For instance, let\[\begin{split}\phi(x,y,z)&=axyz+bxyz^{\prime}+cxy^{ \prime}z+dxy^{\prime}z^{\prime} &+fx^{\prime}yz+gx^{\prime}yz^{\prime}+hx^{\prime}y^{\prime}z+kx^{ \prime}y^{\prime}z^{\prime}=0\end{split} \tag{5}\]be an equation involving three unknown quantities.We can eliminate \(z\) by considering it as the only unknown quantity, and we obtain as resultant\[(axy+cxy^{\prime}+fx^{\prime}y+hx^{\prime}y^{\prime})(bxy+dxy^{\prime}+gx^{ \prime}y+kx^{\prime}y^{\prime})=0\]or\[abxy+cdxy^{\prime}+fgx^{\prime}y+hkx^{\prime}y^{\prime}=0. \tag{6}\]If equation (5) is possible, equation (6) is possible as well; that is, it is verified by some values of \(x\) and \(y\). Accordingly we can eliminate \(y\) from the equation by considering it as the only unknown quantity, and we obtain as resultant\[(abx+fgx^{\prime})(cdx+hkx^{\prime})=0\]or\[abcdx+fghkx^{\prime}=0. \tag{7}\]If equation (5) is possible, equation (7) is also possible;. that is, it is verified by some values of \(x\). Hence we can eliminate \(x\) from it and obtain as the final resultant,\[abcd\cdot fghk=0\]which is a consequence of (5), independent of the unknown quantities. It is evident, by the principle of symmetry, that the same resultant would be obtained if we were to eliminate the unknown quantities in a different order. Moreover this result might have been foreseen, for since we have (SS0.28)\[abcdfghk

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