Question 171887
Let x=9.77777...


{{{x=9.7777}}} Start with the given equation



{{{10x=97.7777}}} Multiply both sides by 10. Notice how when we move the decimal one spot, the numbers after the decimal are EXACTLY the same as the numbers after the decimal above. 



Why is this important? When we go to subtract 9.7777 from 97.7777, the numbers must match in order to completely cancel out the decimal portion.



{{{10x-x=97.7777-x}}} Now subtract "x" from both sides (since we did this to both sides, this does NOT change the equation)



{{{10x-x=97.7777-9.7777}}} Plug in {{{x=9.7777}}} (the original equation) on the right side



{{{9x=88}}} Subtract. Notice how decimal portion of the numbers go away.



{{{x=88/9}}} Divide both sides by 9 to isolate "x"



So the answer is {{{x=88/9}}} which means that {{{88/9=9.77777}}} (where the 7's repeat)