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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

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+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
3a

18α 12ax
30x3x +40=-

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

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4x12 4 6x 6x-4

+
4 3 2 3
4

4
hence &

-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

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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
2a-30

4--4 0
we shall have, x=

0.

2- 1 1
Again, if 4a--b=0, and 2a-3=0; then

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
-2; then
;

4a-6 12--12 2-2 0 20-30 6-6 1-1

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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.

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4

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.. 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 +-
5
-3

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
13-2 21- 2x

to find the value of x.
12
18

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

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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.

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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

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