EQUATION

in algebra, a statement that two expre ions have the same numerical value. An equation may be either identical or conditional. An example of an identical equation, or identity, is (x + y) (w—y) = x2—y*. The left side here can be transformed into the right side, simply by applying the laws of algebra so as to carry out the operations indicated, without taking account in any way of the numerical values of and y. An identical equation is, therefore, true for all values of the variables which appear in it. A conditional equation is not true unle certain special values are a igned to the variables. Thus the equation 4%+7=15 is not true for any value of x except 2. This value 2 ig 206 EQUATION called a root, or solution, of the equation. An equation may have more than one root, e.g. «2+6x”=7 has two roots, 1 and —7; and 273+ aah = 2a + 3 has three roots, 1, -1, Rational Integral Equations — bee Variable). The three equations just given are special cases of the cla of rational integral algebraic equations. The general form of these is axr + bart +... + kx +1=0,where n is a positive integer, and_a, b are given numbers. This equation is said to be of degree n. The branch of mathematics called the Theory of Equations is conventionally restricted to equations of this type. The fundamental result in this subject is that every rational integral equation has a root, a theorem which it is by no means easy to prove. It follows without difficulty that an equation of degree n has exactly n roots, real or imaginary. Two or more of the roots, however, may be equal to each other. To solve an equation is to find its roots. The general equation of degree n can always be solved to any degree of approximation desired, when the numerical values of the coefficients a, b,... are a igned. Graphical methods of solution are often the best (see GRAPH). When the coefficients a, b,... are arbitrary, the general equation can be solved algebraically if m does not exceed 4, but not for greater values of n. It is not that the algebraical solution, or algebraic formula for the roots, when nm is greater than 4, has not been discovered; it does not exist. This was proved more than a hundred years ago by Abel and Galois, two mathematicians of the highest distinction, who both died before they were thirty. For n = 2, the roots of the quadratic equation ax* + be +c=0 pee Ear Ee vie = E00). Manu 3, the cubic equation ax? + ba? + cx +d =0 is reduced to the form 2? + pz+q= Py aM are by putting «=2-— a the solution of 2 + pz +q=0 can be verified to be 2=u-— on where wu is any one of the three cube roots of the quantity — 3a + v(40" + gyp%). Forn = 4, the biquadratic equation is solved with the help of the solution of the cubic. The cubic was first solved by the Italian mathematician Tartaglia, who communicated the solution to Cardan, after binding him to keep it a secret. Cardan, however, gave the solution in his Algebra, published at Nürnberg in 1545. Equations with more than one Variable. A solution of an equation which contains more than one variable is a set of values of the variables making the equation true. Thus the equation «x? — y? = 2 ga solutions (c=5,y = 4), (@= = 0), (x y = —4), and an aniitnited amber of others. When several variables occur, there are usually also several simultaneous equations connecting them; a solution of these is a set of values of 4 variables making all the equations rue. When the number of equation s is equal to the number of variables, there is in general a limited number of solutions of the system. Thus, e.g. the system of equations x? — ye = 2x — y = 6 has two solutions (a = as > y=4), and no A useful rule is that the number of solutions of a system of this type is equal to the product of the degrees of the equations. Exceptions may arise when two solutions coalesce, or when infinite values of the variables occur. Hquations are of great importance in applied mathematics. The data of a problem generally lead to an equation, or a set of equetions, among the quantities concerned. In practice a certain number of these quantities are known in any given case; the unknown quantities are then found by the solution of an equation or equations. Non-algebraic equations occur frequently—equations involving trigonometrical functions, for example. For a modern practical method of solving equations of many types, see NOMOGRAPHY.—BIBLIOGRAPHY: A. HK. Layng, Elementary Algebra; C. Smith, Algebra. More adyanced works are: G. Chrystal, Algebra; W. S. Burnside and A. W. Panton, Theory of Equations.