Let #x and y# be the two numbers.#( x,yinZZ )#
The difference of two numbers is 3.
#i.e . |x-y|=3=>x-y=+-3#
#=>color(red)(x-y=3...to(1) orcolor(blue)( x-y=-3...to(2)#
The difference of their square is 69.
#i.e. x^2-y^2=69#
#=>(x-y)(x+y)=69,#
#where, color(red)(x-y=3# from #(1)or color(blue)(x-y=-3# from #(2)#
#=>3(x+y)=69 or -3(x+y)=69#
#=>color(red)(x+y=23...to(3)) or color(blue)( x+y=-23...to(4)#
Adding #color(red)((1) and (3)) orcolor(blue)( (2) and (4))#
#color(red)(x-y=3)color(white)(............................... color(blue)( x-y=-3#
#color(red)(ul(x+y=23)color(white)(............................. color(blue)(ul(x+y=-23)#
#color(red)(2x=3+23=26)color(white)(................... color(blue)( color(blue)(2x=(-3)+(-23)=-26#
#color(red)(x=26/2)color(white)(.................................. color(blue)( x=-26/2#
#color(red)(x=13)color(white)(.................................... color(blue)(x=-13#
Subst. #color(red)(x=13# in equn. #(3) or color(blue)(x=-13# in equn. #(4)#
we get
#color(red)(13+y=23)color(white)(............................ color(blue)(-13+y=-23#
#=>color(red)(y=23-13)color(white)(....................... color(blue)(y=-23+13#
#=>color(red)(y=10)color(white)(..................................color(blue)(y=-10#
Hence, the reqd. two numbers are:
#color(red)(10 and 13) or color(blue)(-10 and -13 #
