217. There are two other methods of solving Simultaneous Equations of which we have hitherto made no mention, because they are not generally so convenient and simple as the method which we have explained. They are I. The method of Substitution. If we have to solve the equations x+3y= 7 2x+4y=12 we may find the value of x in terms of y from the first equation, thus x=7-3y, and substitute this value for x in the second equation, thus from which we find 2 (7-3y)+4y=12, y=1. We may then find the value of x from one of the original equations. II. The method of Comparison. If we have to solve the equations 5x+2y=16 7x-3y= 5 we may find the values of x in terms of y from each equation, thus Hence, equating these values of x, we get an equation involving only one unknown symbol, from which we obtain y=3, and then the value of x may be found from one of the original equations. 218. If there be three unknown symbols, their values may be found from three independent equations. For from two of the equations a third, which involves only two of the unknown symbols, may be found. And from the remaining equation and one of the others a fourth, containing only the same two unknown symbols, may be found. So from these two equations which involve only two unknown symbols, the value of these symbols may be found, and by substituting these values in one of the original equations the value of the third unknown symbol may be found. Multiplying the first by 7 and the second by 5, we get Subtracting, 35x-42y+28z=105 35x+20y-15≈=95. -62y+43≈=10......... ...(1). Again, multiplying the first of the original equations by 2 and the third by 5, we get from which we can find y=4 and ≈=6. Then substituting these values for y and z in the first equation we find the value of a to be 3. EXAMPLES.-LXXVII. 1. 5x+7y- 2z=13 2. 5x+3y-6z=4 3x-y+22=8 3. 5x-3y+2z=21 4. 4x-5y+2%= 6 2x+3y- z=20 7x-4y+32=35. 5. x+ y+ z = 6 5x+4y+3x=22 · 15x+10y+6z=53. 6. 8x+4y-3x=6 7. x + y + z=30 8x+4y+2x=50 27x+9y+3z=64. 8. 4x-3y+ z = 9 9. 12x+5y-4z=29 10. y-x+z=− 5 z-y-x=-25 XVI. PROBLEMS RESULTING IN SIMUL- 219. IN the Solution of Problems in which we represent two of the numbers sought by unknown symbols, usually x and y, we must obtain two independent equations from the conditions of the question, and then we may obtain the values of the two unknown symbols by one of the processes described in Chapter XV. Ex. If one of two numbers be multiplied by 3 and the other by 4, the sum of the products is 40, and if the former be multiplied by 7 and the latter by 3, the difference between the results is 14. Find the numbers. Let x and y represent the numbers. and then the value of x may be found to be 5. Hence the numbers are 5 and 7. EXAMPLES.-LXXVIII. 1. The sum of two numbers is 28 and their difference is 4, find the numbers. 2. The sum of two numbers is 256 and their difference is 10, find the numbers. 3. The sum of two numbers is 13'5 and their difference is 1, find the numbers. 4. Find two numbers such that the sum of 7 times the greater and 5 times the less may be 332, and the product of their difference into 51 may be 408. 5. Seven years ago the age of a father was four times that of his son, and seven years hence the age of the father will be double that of the son. Find their ages. 6. Find three numbers such that the sum of the first and second shall be 70, of the first and third 80, and of the second and third 90. 7. Three persons A, B and C make a joint contribution which in the whole amounts to £400. Of this sum B contributes twice as much as A and £20 more; and C as much as A and B together. What sum did each contribute? 8. If A gives B 10 shillings, B will have three times as much money as A. If B gives A ten shillings, A will have twice as much money as B. What has each ? 9. The sum of £760 is divided between A, B, C. The shares of A and B together exceed the share of C by £240, and the shares of B and C together exceed the share of A by £360. What is the share of each ? 10. The sum of two numbers divided by 2, gives as a quotient 24, and the difference between them divided by 2, gives as a quotient 17. What are the numbers ? 11. Find two numbers such that when the greater is divided by the less the quotient is 4 and the remainder 3, and when the sum of the two numbers is increased by 38 and the result divided by the greater of the two numbers the quotient is 2 and the remainder 2. 12. Divide the number 144 into three such parts, that when the first is divided by the second the quotient is 3 and the remainder 2, and when the third is divided by the sum of the other two parts, the quotient is 2 and the remainder 6. |
