Question 959788
Looks like you have 4 unknowns and only 3 equations; we will have to use some imagination here
:
Let the digits be a, b, c, d
then
1000a + 100b + 10c + d = "the number"
:
Write an equation for each statement
:
The sum of the digits of a four digit number is 27.
a + b + c + d = 27
:
 The difference of the first two digits is three more than the difference of the last two digits.
a - b = c - d + 3
Rearrange to
a - b - c + d = 3
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 If the digits are reversed, the original number is 459 more than the new number.
1000a + 100b + 10c + d - 459 = 1000d + 100c + 10b + a
Combine like terms on the left
1000a - a + 100b - 10b + 10c - 100c + d - 1000d = 459
999a + 90b - 90c - 999d = 459
simplify, divide by 9
111a + 10b - 10c - 111d = 51
:
Use elimination on the 1st two equations
a + b + c + d = 27
a - b - c + d = 3 
--------------------adding eliminates b and c
2a + 2d = 30
simplify, divide by 2
a + d = 15
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If a + d = 15, then b + c = 12, (the total is 27)
then
a = (15-d)
and
b = (12-c)
Back to this equation
111a + 10b - 10c - 111d = 51
replace a and b
111(15-d) + 10(12-c) - 10c - 11d = 51
1665 - 111d + 120 - 10c - 10c - 111d = 51
combine like terms
-111d - 111d - 10c - 10c = 51 - 1665 - 120
-222d - 20c = -1734
same as
222d + 20c = 1734
Construct a slope/intercept equation and enter into your graphing calc
c = {{{((-222d+1734))/20}}} as y={{{((-222x+1734))/20}}}
Check the table, d=x, c=y; only one positive integer solution comes up
d = 7
c = 9
fInd a and b
a = 15 - 7
a = 8
and
b = 12-9
b = 3
 Find the original four digit number. 8397
:
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See if that checks out using the reverse number difference
8397
7938
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 459
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