Question 919667
let two positive integers be {{{x}}} and {{{y}}}

if the difference of two positive integers is {{{7}}}, then we have:

{{{x-y=7}}} .......eq.1

if their product is {{{42}}},then we have:

{{{x*y=42}}} .......eq.2
 

solve the system:

{{{x-y=7}}} .......eq.1
{{{x*y=42}}} .......eq.2
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{{{x*y=42}}} .......eq.2.......solve for {{{x}}}

{{{x=42/y}}} ........plug it in eq.1


{{{x-y=7}}} .......eq.1

{{{42/y-y=7}}} ....solve for {{{y}}}


{{{42/y=y+7}}}

{{{42=(y+7)y}}}

{{{42=y^2+7y}}}

{{{0=y^2+7y-42}}} ....use quadratic formula to solve for {{{y}}}


{{{y= (-7 +- sqrt( 7^2-4*1*(-42) ))/(2*1) }}}

{{{y= (-7 +- sqrt( 49+168 ))/2 }}}   


{{{y= (-7 +- sqrt( 217 ))/2 }}}

{{{y= (-7 +- 14.73091986265624)/2 }}}


solutions:


{{{y= (-7 + 14.73091986265624)/2 }}}

{{{y= 7.73091986265624/2 }}}

{{{y= 3.865459931328118 }}}

{{{y= 3.87 }}}

or

{{{y= (-7 - 14.73091986265624)/2 }}}

{{{y= -21.73091986265624/2 }}}

{{{y= -10.86545993132812 }}}

{{{y= -10.87 }}}


find {{{x}}}


{{{x=42/y}}}

{{{x=42/3.865459931328118}}}

{{{x=10.86545993132812}}}

{{{x=10.87}}}

or

{{{x=42/y}}}

{{{x=42/-10.86545993132812 }}}

{{{x=-3.865459931328117}}}

{{{x=-3.87}}}


solution pairs: 

{{{x=10.87}}},{{{y= 3.87 }}}

and

{{{x=-3.87}}},{{{y= -10.87 }}}


check eq.1:


{{{x-y=7}}} .......eq.1 for {{{x=10.87}}},{{{y= 3.87 }}}

{{{10.87-3.87=7}}}

{{{7=7}}}

{{{x-y=7}}}......eq.1 for {{{x=-3.87}}},{{{y= -10.87 }}}

{{{-3.87-(-10.87)=7}}}

{{{-3.87+10.87=7}}}

{{{7=7}}}