Question 202020
"The difference of two natural numbers is 9" translates to {{{x-y=9}}}


"the difference of their reciprocals is 9/190" means that {{{1/y-1/x=9/190}}}



Note: if {{{x>y}}}, then {{{1/x<1/y}}}



{{{x-y=9}}} Start with the first equation.



{{{x=9+y}}} Add "y" to both sides.



{{{1/y-1/x=9/190}}} Move onto the second equation



{{{1/y-1/(9+y)=9/190}}} Plug in {{{x=9+y}}}



{{{190*cross(y)(9+y)(1/cross(y))-190y*cross((9+y))(1/cross((9+y)))=cross(190)y(9+y)(9/cross(19))}}} Multiply EVERY term by the LCD {{{190y(9+y)}}} to clear out the fractions.



{{{190(9+y)-190y=9y(9+y)}}} Simplify



{{{1710+190y-190y=81y+9y^2}}} Distribute



{{{1710+190y-190y-81y-9y^2=0}}} Get all terms to the left side.



{{{-9y^2-81y+1710=0}}} Combine like terms.



Notice that the quadratic {{{-9y^2-81y+1710}}} is in the form of {{{Ay^2+By+C}}} where {{{A=-9}}}, {{{B=-81}}}, and {{{C=1710}}}



Let's use the quadratic formula to solve for "y":



{{{y = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{y = (-(-81) +- sqrt( (-81)^2-4(-9)(1710) ))/(2(-9))}}} Plug in  {{{A=-9}}}, {{{B=-81}}}, and {{{C=1710}}}



{{{y = (81 +- sqrt( (-81)^2-4(-9)(1710) ))/(2(-9))}}} Negate {{{-81}}} to get {{{81}}}. 



{{{y = (81 +- sqrt( 6561-4(-9)(1710) ))/(2(-9))}}} Square {{{-81}}} to get {{{6561}}}. 



{{{y = (81 +- sqrt( 6561--61560 ))/(2(-9))}}} Multiply {{{4(-9)(1710)}}} to get {{{-61560}}}



{{{y = (81 +- sqrt( 6561+61560 ))/(2(-9))}}} Rewrite {{{sqrt(6561--61560)}}} as {{{sqrt(6561+61560)}}}



{{{y = (81 +- sqrt( 68121 ))/(2(-9))}}} Add {{{6561}}} to {{{61560}}} to get {{{68121}}}



{{{y = (81 +- sqrt( 68121 ))/(-18)}}} Multiply {{{2}}} and {{{-9}}} to get {{{-18}}}. 



{{{y = (81 +- 261)/(-18)}}} Take the square root of {{{68121}}} to get {{{261}}}. 



{{{y = (81 + 261)/(-18)}}} or {{{y = (81 - 261)/(-18)}}} Break up the expression. 



{{{y = (342)/(-18)}}} or {{{y =  (-180)/(-18)}}} Combine like terms. 



{{{y = -19}}} or {{{y = 10}}} Simplify. 



So the <i>possible</i> solutions for "y" are {{{y = -19}}} or {{{y = 10}}} 

  

However, the problem stated that the numbers are natural. So we have to throw out {{{y = -19}}}



So the only answer for "y" is {{{y=10}}}. This means that "x" is {{{x=9+y=9+10=19}}}



========================================================================================


Answer:



So the solutions are {{{x=19}}} and {{{y=10}}}



So the two numbers are 19 and 10



Take note that {{{19-10=9}}} and {{{1/10-1/19=19/190-10/190=(19-10)/190=9/190}}}