EQUATIONS, GAWIS' THEORY OF.) it was shown to be impossible to solve by radicals any general equation of degree above 4. Subse quently Hermite proved that the roots of the general quintic are expressible in terms of elliptic functions. The quadratic, cubic and quartic are solvable by other methods than those given above, but all are essentially the same. The solution of the general quadratic was known to the Arabs in the 9th century. The solution of x'+ x+q-0 was discovered by Scipio Ferreo in the beginning of the 16th century. It was rediscovered a few years later by Tartaglia. The solution given above is known as Cardan's, but it is known that Cardan learned it from Tartaglia. Ferrari, a pupil of Cardan's, solved the quartic. The solution, given by Bombelli in his algebra (1579), is some times attributed to him. Descartes gave a different solution in 1637. The solution pre sented above is Euler's, having been found by him in 1770.
Higher Equations.- Although the general equations of the 5th and higher degrees are not solvable by radicals, many particular equations of such degrees are thus solvable; e.g., .r"-1) breaks up into two quartics, In works on the theory of equations (see EQUATIONS, THEORY OF) various methods, chief of which is Homer's, are given whereby the commensurable roots of any equation having numerical coefficients can be found and the In commensurable roots can be found to any re quired degree of approximation.
Simultaneous Equations.- The general linear equation in two variables or unknowns, as x and y, is ax+b3=c. Solved for one of the variables, say x, in terms of the other, the equa tion becomes x = It is seen that x and y a are functions of each other: to any value of either corresponds a value of the other. Any two corresponding values constitute a pair satisfy ing the equation. There are infinitely many such pairs satisfying a given equation of the kind in question, as many pairs as there are numbers. Obviously there are hosts of pairs not satisfying, a given equation. All the pairs satisfying a given equation constitute a systrii of pairs. Two equations tax biy= different unless at : as= los: bit= ct.
Have the two systems determined by two dif ferent equations any pairs in the common? The answer is, one pair. It can be found as fol lows : Multiplying the former equation by Is, the latter by - adding and solving for x, .r= (b2ct- bici) : aibi); analogously, y (aici - cia,) : °A). This and only this pair of values of x and y satisfies both equa tions. In combining the equations, x and y were regarded as the same in both. Two or more equations in two or more unknowns are called simultaneous when the unknowns are treated as representing the same numbers in aD the equations. In the foregoing solution the x-equation was found eliminating y bettveti the given equations. The elimination was ac complished by addition. It might have been done otherwise, as by comparison, i.e., solving each equation for y and equating the y-values so obtained, or by substitution, i.e., solving one of the equations for y and substituting the e value so found for y in the other equation. In any of these ways or by combinations of them one may find a triplet of values satisfying three arbitrary equations in three unknowns, r. y s: eliminate, say s, between two of them and then between the remaining one and one of the others; so result two equations in x and y, to be handled as above. The method is obviously extensible to the use of is equations in n unknowns. In general, is linear equations in is unknowns are sat isfied simultaneously by but a single set of values, but in special cases by no se or by more than one set. The latter happens only when the coefficients satisfy some special condition or conditions. Under certain conditions is or more equations in is -1 un knowns may be satisfied by the same set of value. Thus ax b = 0 and cx d = 0 hail the same root when and only when be — ad =0: sax biy cz= 0, ass = 0, ch.r I,y + 4=0, are simultaneously satisfied or are consistent when and only when a,b,c. abort + (h&c.= 0: For the ex pression of such conditions and the solution of sets of linear of deter minants, see the article IILDIANTS and works therein cited.