Question 120464
5. let x=the number to be added, then we have:

(7+x)/(9+x)=5/7
7+x=5
x=-2
or
9+x=7
x=-2
another way:
Multiply each side by 7(9+x)---the LCM
7(7+x)=5(9+x) get rid of parens

49+7x=45+5x  subtract 7x and also 45 from both sides
49-45+7x-7x=5x-7x collect like terms

4=-2x divide both sides by -2
x=-2

CK
(7-2)/(9-2)=5/7
5/7=5/7

7.

Let x=one even integer; reciprocal=1/x
and (x+2)=the other; reciprocal=1/(x+2)

Now we are told that:

(1/x)+1/(x+2)=5/12  Multiply each term by 12(x)(x+2)

12(x+2)+12x=5x(x+2)  get rid of parens
12x+24+12x=5x^2+10x simplify
24x+24=5x^2+10x  subtract 24x and also 24 from both sides
24x-24x+24-24=5x^2+10x-24x-24  collect like terms

5x^2-14x-24=0  quadratic in standard form---solve using the quadratic formula:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
{{{x = (14 +- sqrt( 14^2-4*5*(-24) ))/(2*5) }}} 
{{{x = (14 +- sqrt( 196+480 ))/(10) }}} 
{{{x = (14 +- sqrt( 676 ))/(10) }}} 
{{{x = (14 +-26 )/(10) }}} 
{{{x = (40)/(10)=4 }}}---------------------------------One integer
{{{x+2=4+2=6}}}-----------------------------------------The other integer
and
{{{x = (-12)/(10) }}}----------------------Not an integer 

CK
1/4+1/6=5/12
3/12+2/12=5/12
5/12=5/12

Hope this helps---ptaylor