If the graphs of two equations do not intersect, these equations express conditions that cannot be true at the same time and hence the equations are not true simultaneous equations. 92. From what has been said so far, it is clear that a graph shows how two variables change in value. Hence, such a diagram may be used whenever we wish to show the variation of two related quantities. A common instance is a temperature graph, showing the variation in temperature as the time varies. In this case, horizontal units represent units of time and vertical units represent degrees of temperature. See Fig. VII. Let each horizontal unit represent 1 hour and each vertical unit 5 degrees of temperature. The line AB shows how the temperature varies from 2'oclock A.M. to 10 P.M. At any time of day, the vertical distance shows the temperature at that time. Thus at 2 A.M. the temperature was 60°, while at noon, the temperature had risen to about 87°. After reaching a maximum of almost 90° shortly before 3 o'clock P.M. the temperature fell steadily until at 10 A.M. it was again down to 60°. The student should have no difficulty in finding similar graphs in current magazines. CHAPTER XII SIMULTANEOUS SIMPLE EQUATIONS 93. As has been shown in the previous chapter, simple simultaneous equations may be solved graphically by finding a point common to the two graphs. The algebraic solution also depends upon a combination of the two equations, such, that one of the two unknown numbers may be made to disappear, leaving an equation, containing only one unknown, which may be solved by the usual method. Elimination is the name given to the process of making one of the unknown numbers disappear. There are three general methods for eliminating one of the unknowns. Since the terms containing y in equations and have equal coefficients but opposite signs, they will cancel each other when the equations are added. Therefore adding and 3, 7x = 35 To eliminate y, their coefficients must be made equal. Since the y terms have like signs, we eliminate by subtraction. Subtract from, TO ELIMINATE BY ADDITION OR SUBTRACTION Rule.-Multiply the equations by numbers that will make the coefficients of one of the unknowns numerically equal. Eliminate this unknown by adding the resulting equations if the equal coefficients have unlike signs, or by subtraction if their signs are alike. It is usual to select that unknown for elimination which will require the smallest multipliers. These multipliers can readily be found if the student will note that the equal coefficients will be the least common multiple of the two original coefficients. Thus in the preceding problem, the L. C. M. of the coefficients of y is 6. Finding the value of y in terms of x in equation 1, This equation contains only one unknown, x, and can therefore be solved in the usual way TO ELIMINATE BY SUBSTITUTION Rule. Find the value of one of the unknown numbers, in terms of the other, in one of the equations and substitute this value in the other equation. |