Thread: SAT problem

the question is:
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+y+z
2. w=Y+1
3. Z=W-5
what is the four-digit integer?

I have tried every system of equation and substitution I can think of!

Obviously, you have four variables and three equations, so you will not be able to find a unique solution using regular methods for solving systems of equations. You have to use the facts that digits are integers between 0 and 9.

We have x = 3(y - 1) <= 9. On the other hand, z = y - 4 >= 0, so y >= 4. This gives a unique value for y, from which other digits can be found.

In mathematics (as well as in physics and programming), it's not good to denote the same variable by a lower- and uppercase letters.

Originally Posted by klg

the question is:
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+y+z
2. w=Y+1
3. Z=W-5
what is the four-digit integer?

I have tried every system of equation and substitution I can think of!

w = y+1
z = w-5 = (y+1) - 5 = y-4 ... this says y > 4

x = w+y+z = 3y-3 using the above equations

x = 3y-3 implies that y has to be one of {1,2,3,4} ... look two lines up for the additional restriction.


you should be able to figure it out from here ...

Got it.
Thanks!

Hello, klg!

This problem doesn't require fancy algebra . . .

A four-digit integer, WXYZ, in which W,X,Y and Z each represent a different digit,
is formed according to the following rules.

. .
. .
. .

What is the four-digit integer?

From [2]: .
From [3]: .

Substitute into [1]: .


We have: .


Since , we have: .

Since , we have: .

Therefore: .

That is WONDERFULLY simple, and seems so clear now!
Thank you so much!
klg