The difference of two numbers is 3 and the difference of their square is 69. What are the numbers?

1 Answer
May 8, 2018

Answer:

#color(red)(10 and 13) or color(blue)(-10 and -13 #

Explanation:

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 #