Question 132604
First number: {{{x}}}
Second number: {{{y}}}


The two numbers differ by 6: {{{x-y=6}}}  We'll just assume that {{{x>y}}}


The reciprocal of the first number: {{{1/x}}}, {{{x<>0}}} and because of {{{x-y=6}}}, {{{y<>-6}}}
The reciprocal of the second number: {{{1/y}}}, {{{y<>0}}} and because of {{{x-y=6}}}, {{{x<>6}}}



Their reciprocals differ by 8/15:  Since we have assumed that {{{x>y}}}, {{{1/x<1/y}}}, so {{{1/y-1/x=8/15}}}


Re-write the first equation to solve it for x:
{{{x=y+6}}}


Substitute this expression for x into the second equation in place of x.
{{{1/y-1/(y+6)=8/15}}}


Multiply both sides by 15:
{{{15/y-15/(y+6)=8}}}


Lowest common denominator is just the product of the two denominators since there are no common factors.  So:


{{{(15(y+6)-15y)/(y(y+6))=(8y(y+6))/(y(y+6))}}}


Distribute and collect like terms all on the left:
{{{(-8y^2-48y+15y+90-15y)/(y(y+6))}}}
{{{(-8y^2-48y+90)/(y(y+6))}}}


Solve the numerator expression for 0, remembering to exclude 0 and -6 as values not in the domain of the function in terms of y.


{{{-8y^2-48y+90=0}}}


Divide by -2:
{{{4y^2+24y-45=0}}}


Factor:
{{{(2x-3)(2x+15)=0}}}, so y = 1.5 or y = -7.5


y = 1.5 => x = 7.5 because {{{7.5-1.5=6}}}, so:


{{{1/1.5-1/7.5=5/7.5-1/7.5=4/7.5=8/15}}} and the first root checks.


y = -7.5 => x = -1.5 because {{{-1.5-(-7.5)=6}}}, so:


{{{1/(-7.5)-1/(-1.5)=-1/7.5+5/7.5=4/7.5=8/15}}} and the second root checks.