International projectionist (Jan-Dec 1939)

The Fundamentals of Mathematics if fl = H, . ■. fc = 1J< a = 1, . ••6 = 2A a = 2, . :b = -x a = 3, . : b 2 By GEORGE LOGAN SOUND DEPARTMENT, METRO-GOLDWYN-MAYER STUDIOS /F. Simultaneous Equations. It will be a help if the reader digest each article as it appears, for the ideas presented in subsequent sections hinge upon an understanding of topics discussed in earlier sections. Further, it is desirable that the issues of this series be cached away after reading, as back-reference may be useful before the series is completed. IT IS worth while to make a study of the methods used in solving simultaneous equations, for in the methods to be discussed we must necessarily use most of the various fundamental operations which have been established in the three previous sections of this series. In brief, we'll get a good review in the handling of equations. Also, they often crop up in practical problems of mechanics and electricity. As a case in point, a simultaneous equation solution of a branched circuit is included at the end of this article. Simultaneous equations are two or more independent equations involving the same unknowns. Thus, if a and b are the unknowns, two simultaneous equations involving these unknowns could be: 4a+36 = 6 2a 6 = 4 It is evident that these equations are independent, because they express different relations of the unknowns. But suppose we have: 4a+36 = 6 8a+66 = 12 These equations, it is apparent, are not independent, for if we multiply both sides of the first equation by 2, the second equation is obtained. Hence the second equation can be reduced to identity with the first equation simply by dividing both sides of the second equation by 2; and therefore the second equation is not independent of the first equation. If we have but one equation in two unknowns, such as: 4a+36 = 6 it is possible to find an infinite number of values for each of the two unknowns which will make the equation true; that is, satisfy it. This can be easily shown by arbitrarily setting values for one unknown and then computing the other. If in the equation immediately above we assign various values to a: and so on indefinitely. But if we have two simultaneous equations involving two unknowns, there is possible only one value for each of the unknowns. In other words, if we have two simultaneous equations involving a and b, there is only one value for a and only one value for b which will satisfy both equations. When that one possible value for each of the unknowns is found the simultaneous equations are solved. To reach the solution operations are performed on the equations so that a single equation is derived from them involving just one unknown. This process is called elimination, quite naturally, for in this single derived equation all but one unknown are eliminated. Any one of several means of elimination may be used, according to which means is most conveniently applied to a given problem. The several means of elimination may be listed: 1. By addition or subtraction of simultaneous equations. 2. By substitution of values. 3. By equating equal expressions. Each of these operations have the common goal of making all but one unknown disappear, so that a numerical value can be found for that one unknown. Let us consider the first method listed. 1. Elimination by addition or subtraction of simultaneous equations. Assume that we have the appended simultaneous equations Nos. 1 and 2. 1. 4x+3y = 6 2. 4x-3.v = 18 Inspection shows that if we add Nos. 1 and 2 the resultant equation, No. 3, will not have a y term. 1. 4x+3y = 6 2. 4x-3.v = 18 3. 8x = 24 Solve for x from No. 3: 3. 8x = 24 x = 24 8 x — 3 Thus from addition of our simultaneous equations we eliminate one of the unknowns, y, and obtain a value for the other unknown, x. Now to find the value of y, simply place x = 3 in either No. 1 or No. 2. 1. 4x+3y = 6 (4)(3)+3y = 6 12+3y = 6 3y = 6-12 3y = -6 y = -2 Hence the solution of Nos. 1 and 2 is x = 3 and y = — 2. To check for correctness, place these determined values in No. 2: 2. 4x-3y = 18 (4)(3)-(3)(-2) = 18 12+ 6 = 18 18 = 18 check We can take this same problem and solve it by subtracting the simultaneous equations. Subtract equation No. 2 from equation No. 1: 1. 4x+3y = 6 2. 4x-3y = 18 6y = -12 y = -2 Place y = —2 in equation No. 1: 1. 4x+3y = 6 4x+(3)(-2) = 6 4x-6 = 6 4x = 12 x = 3 It is apparent that to eliminate an unknown through addition or subtraction of simultaneous equations, the coefficient for that unknown must be numerically the same in each equation. Equation Nos. 1 and 2 comply with this requirement as they stand — that is, the coefficient of x is numerically 4, and the coefficient of y is numerically 3, in both equations. Discretion as to whether we use addition or subtraction depends on the signs of the equal coefficients. If in two equations the coefficients of an unknown are equal and of opposite sign, addition will make the unknown disappear. If in two equations the coefficients of an unknown are equal and of the same sign, subtraction will eliminate the unknown. We know that we can multiply both sides of any equation by any chosen 12 INTERN ATION AL PROJECTIONIST