by transposing - 36 and 2bx, it becomes 3ax-2bx=36, by collecting the coefficients of x, we shall have (30—2b)=3b, 36 by division, 3a - 26 a +4a, 2 *200. Now, if in this example, we suppose 3a - 2= 36 0,0r 3a=2b, then x=0; which shows that in the 0 above equality such a relation cannot exist between the quantities a and b, or if there should, the equality cannot take place. Let us, in order to see what would be the result, , 3a substitute for b in equation (1), and it becomes 2 18α 12ax 2 2 multiplying by 2, we shall have. 6ax-9a +-8a=18ax -12ax + 8a, by transposition, 18ax-18ax=9a, ..0=9a. Which is evidently absurd, in all cases, except that a=0. and therefore b=0, and then the original equation is nothing else than 0=0 in its primitive state, We may therefore conclude that there is no finite value, which, when substituted for x in the primitive equation, would fulfil the condition required, this may be better, verified by a numerical example, Thus, let a=4, and b=6; then substituting these values for a and 6 in the given equation, it becomes 4x12 4 6x 6x-4 + 4 -3+ -=3x - 2rti.. 3=0. 3 3 Es. 11. Given 2ax+b=3cx+4a, to find the value of . by transposition, 2ax—3cx=4a-b, hy collecting the coefficients, (20—30)x=40-6, 40-6 .. by división, x= 20-30 201. Tiere, if 4a=b, and at the same time, 2a> or <3c; then x=0. For a=1,b=4 and c=}, then, the above equation becomes 23 -x=4-4, .'.x=0. Or, substituting these values of a, b, and c, in the formula, 40-b 4--4 0 0. 2- 1 1 4a-b 0 ; 20-30 0 which is the mark of indetermination, or, which is the same thing, we learn from this result, that the value of x may be any number, either positive or negative, from nought to infinity, and both inclusively. In order to illustrate this, let a=3,b=12, and 4a-6 12--12 2-2 0 20-30 6-6 1-1 3 Now, let us resume the given equation, and by substituting these values of a, b, and c, we shall have 6x +12=6x+12,... 6x=62. But this is what Analysts call an identical equation; where it evidently appears that w is indeter 2 minate, or that any quantity whatever may be substituted for it. Ex. 12. Given 19x+13=59-4r, to find the value of x. by transposition, 19.c +4x=59-13, or, 23x=46; .. by division, x=2. Ex. 13. Given 3x +4-=46-20, to find the , value of x. Multiplying both sides by 3, 9x+12-x=138—6x, by transposition, 9x+6x-x=138-12, or 14x=126 ; 126 by division, I= 14 Ex. 14. Given x2 +15x=35x-3x2, to find the value of x. Dividing every term by x, x +15=35-3x, by transposition, x+3x=35-15, or 4x=20; ..x=5. Ex. 15. Given +10= +11, to find the 6 3 2 value of x. Here 12 is the least common multiple of 6, 4, 3, and 2; x=9. 4 + .. multiplying both sides of the equation by 12, 2x -3x+120=41-62 +132; by transposition, 2x - 30—4x+6x=132-120, or 83-7x=12; ..x=12. - 1 23 4 Ex. 16. Given 4+*, to find 7 5 4 the value of x. Ans. x=8. 7.2 +5 16+43 3x +9 Ex. 17. Given +6= to 3 5 2 find the value of x. Ans. x=1. 17-32 4x + 2 7x +14 Ex. 18. Given =5—6x +- 3. to find the value of x. Ans. x=4. 3x -3 20-3 6x —8 Ex. 19. Given x +4= + 5 2 7 4x to find the value of x. Ans. x=6. 5 4x - 21 57-32 Ex. 20. Given 十3十一 = 2419 4. 50-96 11x, to find the value of x. 12 Ans. x=21. 6x +18 11-23 Ex. 21. Given -45 =54-48 9 36 to find the value of x. Ans. x=10. a2 -33 6bx - 5a2 Ex. 22. Given ax -abs =bx+ 2a bx+4a to find the value of x. 4 4ab2 - 100 Ans. X= 40-36 700 +18 +8 Ex. 23. Given to find the 21 42-11 3' value of x. Ans. x=8. 6x +7, 78—13 20+4 Ex. 24. Given + to find 9 6x +3 3 the value of x. Ans. x=4. 4x +3, 7-29 8x +19 Ex. 25. Given + to find 9 5x 12 18 the value of x. Ans. x=6. Ex. 27. Giyen 50 Ex. 26. Given 12-X: :: 4:1, to find the va 2 lue of x. Ans. x=4. 500 +4 18 ::7:4, to find the 2 4 value of x. Ans. x=2. # Ex. 28. Given (2x+3) = 4x +14x +172, to find the value of x. Ans. x=6. 3x +4 22Ex. 29. Given +- 2c +16, to find 5 5 the value of x. Ans. x=7. 7 33 - 11 8x +15 Ex. 30. Given + to 2 4 6 find the value of x. Ans. x=3. x2 Зах 2 Ex. 31. Given to find the value 2 2 2 1 Ans. x = 3a-1 Ex. 32. Given 20C x +3 12x +26 +15= to find 3 5 the value of. x. Ans. x=12. Ex. 33. Given 5ax --26 +46x=2x +- 5c, to find 5c+2b the value of x. 5a +46-2 +4 + of x. Ans. x |