54.

(a + b −c)2 - d2 (b + c − a)2 - d2
+

+

(c + a−b)2 — d2

(a + b)2 − (c + d)2 1 (b + c)2 − (a + d)2 + (c+a)2 − (b+d)2°

61. If y+z

prove

C - -d

c2 d2

c2 + d2

a2 - b2

(iii.)

=

C-
- 2 d

a2 + b2

(iv.)

pa2 + qab + rb2

la2 + mab + nb2

pc2+qcd+rd2 Ic2+mcd+nd2

=

c2

(c + d) 2

Y =

2

; then will x + y + z = 0.

=

ay+bz

2+x x+y
; then will each fraction =
az + bx ax+by

2 a+b

62. Prove that, if a, b, c be unequal, and

then will x + y + z = 0 and ax + by + cz = 0.

CHAPTER XV.

EQUATIONS WITH FRACTIONS.

171. In the present chapter we shall give examples of equations which contain fractional expressions.

We may multiply both sides of the equation by 120, the L. C. M. of the denominators of the fractions, without destroying the equality; we thus get rid of fractions, and have

24(x − 1) — 15 (3 x − 1) = 20 (43 − 5 x) ;

The L. C. M. of the denominators is (x − 3) (x + 3) (x + 5). Multiplying both sides of the equation by the L. C. M.,* * we get rid of fractions, and have

(x+3)(x+5)+ 2 (x − 3) (x+5)=3(x-3) (x+3);

... x2 + 8x + 15 +

2x2 + 4x 303x2 - 27.

*See Art. 181.

In this case it is best not to multiply at once by the L. C. M. of the denominators; the work is simplified by proceeding as under.

which is the same equation as that in Ex. 3.

x-2

х

X

1.

- 7

5

10

3x

1

1

2.