Finding the values of x the equations, 20 x = 7 14 + 4y Placing these two values equal to each other, 14 + 4y x = 7 in terms of y, from each of 20 6y = Combining (1) and ting z in each case, by RULE. has been eliminated by comparison. Here, In the same way, an unknown quantity may be elimɩ nated between any two equations. Hence, the Find from each equation the value of the quantity to be eliminated; place these values equal to each other. Of the three rules given, either one can be used, as may be most convenient. As a general rule, that one will be employed which gives rise to the simplest equa tions. 10 2y+32 16 4x + 2y + 2z = 22 SOLUTION OF GROUPS OF SIMULTANEOUS EQUATIONS. 94. Take the three equations, 3x + 4y 22 5x 19x8y = 62 (2), also (1) and (3), elimina we have the new group, (1.) (2.) (3.) (4.) (5.) Combining (4) and (5), eliminating y, single equation, 29x58 .'. x = 2. 6 + 12 Substituting this value of x in (5), we have, 14+6y= 32 ..y = 3. Substituting these values of x and y in (1), we have, we have the In the same way, any group of simultaneous equations may be solved. Hence, the RULE. I. Combine one equation of the group with each of the others, eliminating one unknown quantity: there will result a new group containing one equation less than the original group. II. Combine one equation of this group with each of the others, eliminating a second unknown quantity: there will result a new group containing two equations less than the original group. III. Continue the operation until a single equation is found, containing but one unknown quantity. IV. Find the value of this unknown quantity by the preceding rules; substitute this in either one of the group of two equations, and find the value of a second unknown quantity; substitute these in either of the group of three, finding a third unknown quantity; and so on, till the values of all are found. In making the combinations, care should be taken to make them in such a way as to obtain as simple equa tions as possible. When any unknown quantity does not enter all of the equations, it will in general be best to eliminate that quantity first. Combining (2) and (4), And by successive substitution, 4. 5. B. Solve the following groups of simultaneous equations 3x + 4y 2x y = 7. 10. 11. 9. x 5x + 3y =.26 5x ― 4x + 3y = 16 3x + 4y = 19 8. 4x+5y= 17 3y-2x= 8 y 6x + y = 12 x + by = 37 - = 18 1 3y = 6 2x + 9y= 17 f = 2 and 2 = 4. - x + y + z = 6 5x + 2y 3z = 0 = 1 792 158 2 2. y 9 x + y = .. 2 = Ans. Ans. Ans. |