Question 431489
Call the numbers {{{a}}} and {{{b}}}
given:
(1) {{{ a = b + 3 }}}
(2) {{{ 1/a + 1/b = 1/2 }}}
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Multiply both sides of (2) by {{{2a*b}}}
(2) {{{ 2b + 2a = a*b }}}
By substitution:
(2) {{{ 2b + 2*(b + 3) = (b + 3)*b }}}
(2) {{{ 2b + 2b + 6 = b^2 + 3b }}}
(2) {{{ b^2 - b = 6 }}}
Complete the square
{{{ b^2 - b + (1/2)^2 = 6 + (1/2)^2 }}}
{{{ ( b - 1/2)^2 = 24/4 + 1/4 }}}
{{{ ( b - 1/2)^2 = 25/4 }}}
Take the square root of both sides
{{{ b - 1/2 = 5/2 }}}
{{{ b = 3 }}} and also
{{{ b - 1/2 = -5/2 }}}
{{{ b = -2 }}}
And, from (1),
(1) {{{ a = -2 + 3 }}}
{{{ a = 1 }}}
The numbers are 1 and -2
check answer:
(2) {{{ 1/a + 1/b = 1/2 }}}
(2) {{{ 1/1 + 1/(-2) = 1/2 }}}
{{{ 1 - 1/2 = 1/2 }}}
OK