SOLUTION: A four-digit integer WXYZ, in which W, X, Y, and Z each represent a different digit, is formed according to the following rules 1.) X = W + Z + Y 2.) W = Y +1 3.) Z = W

SOLUTION: A four-digit integer WXYZ, in which W, X, Y, and Z each represent a different digit, is formed according to the following rules 1.) X = W + Z + Y 2.) W = Y +1 3.) Z = W - 5

Question 771430: A four-digit integer WXYZ, in which W, X, Y, and Z each represent a different digit, is formed according to the following rules
1.) X = W + Z + Y
2.) W = Y +1
3.) Z = W - 5
What is the four-digit integer?
I tried plugging in, say, Y + 1 for W in the first equation, then W - 5 for Z and Y + 1 for W, trying this method in the other equations, but I haven't gotten anywhere with this.
Could you please help me find a way to solve this? If it helps, the answer is 5940. I'm practicing for the SAT coming up.
Thank you!
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
A method that worked was use the w formula in the other two equations to eliminate it through substitution:

Leaving out some of the steps,
x-2y-z=1 and z-y=-4

Next, let , some as yet not known constant,
and putting into a well chosen form,
and
then using and

That looks like an intermediary form for y and so for now, we have:
-------------------

------------------

We continue to solve for x.
Substitute what we found for y into x-2y=k+1,

So we have this list of somewhat solved variables:
--------------------

--------------------

Now, return to the formula specified for w.
, our given formula of w,
, substituted for y,

Now our summary of nearly solved digits (variables) is this:
-----------------------

----------------------

SO WHAT IS k FOR FINDING EACH OF THE DIGITS?
Our smallest digit can be at least 0 and our largest digit can be at most, 9.
Look at x. Already with a constant 9 in it. Let k=0, so the term there does not contribute anything. The result is that:
ANSWER FOUND__________w=5, x=9, y=4, z=0.
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