Hence, multiplying each side by the L.C.M. of the denominators

63. When each side of an equation consists solely of a single fraction, the numerator of either fraction may change places ·with the denominator of the other.

Multiply each side by b, then, by Art. 60 (1.)—

Divide each side by p, then, by Art. 60 (2.)

Here the denominator b of the first side of the given equation has changed places with the numerator p of the second side.

=

where the other

α

And similarly we may show that numerator and denominator have changed places.

α

COR. The two sides of an equation of the form of

be inverted.

b Չ

For interchanging p and b in the last result, viz.,

and therefore also, by Art. 60 (5.), we have

Ρ

(The student is cautioned against inverting the separate terms of the two sides of an equation when there are more than one term on each side.)

64. When each side of an equation consists solely of a single fraction, we may perform the following operations:

1. We may add or subtract the numerator and denominator of EACH fraction for a new numerator or denominator, and retain either the original numerator or denominator for the other term of the fraction, both sides being always similarly treated.

or, (v.) we may have equations formed by inverting each of

And so, by subtracting unity from each side, we get—

2. We may take the SUMS of the numerator and denominator of each for new numerators or denominators, and the DIFFERENCES for the other terms of the fraction; and VICE VERSA, both sides being always similarly treated.

and

α b

-

a + b

a + b

a

=

=

c + ď

P+ !
q

? - {
4.

i

And inverting each side, we have, by Art. 63 (Cor.)—

Then, by Art. 64 (1.), retaining the numerators and taking the differences for new denominators, we have

We may consider the quantity a as a fraction whose deno

Then, Art. 64 (2.), taking the sum and difference, we have

Again, taking the difference and sum, we have

65. We now give an example to show that sometimes the easy solution depends on an advantageous arrangement of the terms on the two sides of the equation.

Should the student commence by squaring at once, he will render the equation more complicated.