Question 257190
let a be the larger number.
let b be the smaller number.


formulas are:


{{{sqrt(a) + sqrt(b) = 5}}} (first equation)


{{{a-b = 5}}} (second equation)


use the first equation to solve for a in terms of b.


first equation is:


{{{sqrt(a) + sqrt(b) = 5}}}


subtract {{{sqrt(b)}}} from both sides of this equation to get:


{{{sqrt(a) = 5 - sqrt(b)}}}


square both sides of this equation to get:


{{{a = 25 - 10*sqrt(b) + b}}}


substitute in second equation to get:


{{{25 - 10*sqrt(b) + b - b = 5}}}


combine like terms to get:


{{{25 - 10*sqrt(b) = 5}}}


subtract 25 from both sides of this equation to get:


{{{-10*sqrt(b) = 5-25 = -20}}}


divide both sides of this equation by (-10) to get:


{{{sqrt(b) = 2}}}


square both sides of this equation to get:


{{{b = 4}}}


if b = 4, than a-b = 5 becomes a-4 = 5 becomes a = 9.


you have:


a = 9
b = 4


a-b = 5 becomes 9-4 = 5 confirming second equation is true.


{{{sqrt(a) + sqrt(b) = 5}}} becomes {{{sqrt(9) + sqrt(4) = 5}}} becomes 3+2 = 5 confirming second equation is true.


answer is:


the numbers are 4 and 9.