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[ G.Polya ] Setting up equations is like translation from one language into another ( NOTATION,1 ). This comparison, used by Newton in his Arithmatica Univeresalis may help to clarify the nature of certain difficulties often felt both by students and by teachers. 1. To set up equations means to express in mathematical symbols a condition that is stated in words; it is translation from ordinary language into the language of mathematical formulas. The difficulties which we may have in setting up equations are difficulties of translation In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. first we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression. An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idiom, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the whole meaning; before translating the sentence, we may have to rearrange it. It is very much the same in setting up equations. In easy cases, the verbal statement splits almost automatic ally into successive parts, each of which can be immediately written down in mathematical symbols. In more difficult cases, the condition has parts which cannot be immediately translated into mathematical symbols if this is so , we must pay less attention to the verbal statement, and concentrate more upon the meaning. Before we start writing formulas, we may have to rearrange the condition, and we should keep an eye on the resources of mathematical notation while doing so. In all cases, way or difficult, we have to understand the condition, to Separate the various parts of the condition and to ask Can you write them down , In easy cases we succeed without hesitation in dividing the condition into parts that can be written in mathematical symbols in difficult cases, the appropriate division of the condition is less obvious. The foregoing explanation should be read again after the study of the following examples 2. Find two quantities who's' sum is 78 and who's product is 1296. We divide the page by a vertical line. On one side, w e write the verbal statement split into appropriate parts. on the other side, we write algebraic signs, opposite to the corresponding part of the verbal statement. The original is on the left, the translation into symbols on the right. Stating the problem in English in algebraic language Find two quantities x y who's sum is 78 and x + y = 78 whose product is 1296 xy=1296 In this case, the verbal statement splits almost automatically into successive pars, each of which can be immediately written down in mathematical symbols 3. find the breadth and the height of a right prism with a square base, being given the volume, 63 cu in and the area of the surface 102 sq in what are the unknown? the die of the base, say x and the altitude of the prism, say y. What are the data? the volume 63, and the area, 102. What is the condition? the prism whose base is a square with side x and whose altitude is y must have the volume 63 and the are a02. Separate the various parts of the condition. There are two parts, one concerned with volume, the other with the area. We can scarcely hesitate in dividing the whole condition just in these two parts; but we cannot write down these parts "immediately" We must know how to calculate the volume and the various parts of the area. yet if we know that much geometry, we can easily restate both parts of the condition so that the translation into equations is feasible. We write on the left hand side of the page an essentially rearranged and expanded statement of th problem, ready for translation into algebraic language Of a right prism with a square base find the side of the base x and the altitude y First the volume is given 63 the are of the base which is a square with side x^2 and the altitude y determine the volume which is their product x^2y=63 Second, the area of the surface is given 102 the surface consists of two squares with side x 2x^2 and of four rectangles each with base x and altitude y 4 xy and whose sum is the area 2x^2 + 4xy = 102 Coordinates of a point, find the point which is symmetrical to the given point with respect to the given straight line. this is a problem of plane analytic geometry. what is the unknown? a point, which coordinates, say p,q. What is given? the equation of a straight line, say y= mx+n, and a point with the coordinates, say a,b. What is the condition? the points a(b, and p,q) are symmetrical to each other with respect to the line y=mx+n. we now reach the essential difficulty which is to divide the condition into parts each of which can be expressed in the language of analytic geometry. The nature of this difficulty must be well understood. A decomposition of the condition into parts may be logically unobjectionable able and nevertheless useless. what we need here is a decomposition into parts which are fit for analytic expression. in order to find such a decomposition, we must go back to the definition of symmetry but keep an eye on the resources of analytic geometry what is meant by symmetry with respect tho what is meant by symmetry with respect to a straight line ? What geometric relations can we express simply in analytic geometry? we concentrate upon the first question but we should not forget the second. thus, eventually we may find the decomposition which we are going to state. The given point (a,b) and the point required (p,q) are so related that first the line joining them is perpendicular (q-b)/(p-a) = -1/m to the given line and second the midpoint of the line joining them lies on the given line. p+q/2 = m a+p/2 +n