by transposing-36 and 2bx, it becomes 3ax-2bx=3b, by collecting the coefficients of x, we shall have (3α-2b)=3b, by division, x 3b 3a -26 $200. Now, if in this example, we suppose 3a-2b= 36 0,or 3a=26, then x; which shows that in the 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 6 in equation (1), and it becomes 2 multiplying by 2, we shall have • 6ax-9a+8a-18ax-12ax+8a, by transposition, 18ax-18ax-Ja, ...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 6-6; then substituting these b=6; values for a and b in the given equation, it becomes Ex. 11. Given 2ax+b=3cx+4a, to find the va lue of x. by transposition, 2ax-3cx-4a-b, by collecting the coefficients, (2a-3c)x-4a-b, 2a-3c 201. Here, if 4a=b, and at the same time, 2a> or <3c; then x=0. For a=1,b=4 and c=1, then, the above equation becomes 2x-x-4-4, .. x= 0. Or, substituting these values of a, b, and c, in the formula, Again, if 4a-b-0, and 2a-3c=0; then 4a-b 0 2a-3c 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 =2; then 30: 4a-b 12-12 2-2 0 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=6x. But this is what Analysts call an identical equation; where it evidently appears that x is indeterminate, or that any quantity whatever may be substituted for it. Ex. 12. Given 19x+13-59-4x, to find the value of x. by transposition, 19x+4x=59-13, Ex. 13. Given 3x+4- =46—2x, to find the value of x. 3 Multiplying both sides by 3, 9x+12-x=138-6x, by transposition, 9x+6x-x-138-12, 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=3+11, to find the value of x. x 6 4 Here 12 is the least common multiple of 6, 4, 3, and 2; |
