Help me with this SAT Math problem!?

Question

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?

explanation as well? because i want to learn how to do it as well. thank you for your taking your time to solve this question.

Answer 1

x = w+y+z
and w=y+1 , z = w-5
Therefore,
x = w+y+z
x = (y+1)+y+[(y+1)-5]
x = 3y - 3
We know x will be greatest one digit number.
Therefore, if we take x as 9.
we will get
x = 3y -3
9+3 = 3y
y = 4.
And,
w = y+1
w = 4+1
w = 5
also,
z = w-5
z = 5 - 5
z = 0
W = 5, X=9, Y = 4 , Z = 0
WXYZ = 5940

Answer 2

X = W + W-1 + W-5 = 3W - 6.

Since X <=10, 3W <=16, and therefore W <=5 (since it's a whole number).

But by #3, since Z >=0, W>=5.

I.e., W = 5. So Y=4, Z=0, X=9.

The fact that they're all different wasn't even needed.

Also not needed was trial and error. It's a legitimate tactic for problems of this general kind, but it is NOT needed in this particular case.

Answer 3

From (3) W > = 5 or Z will be negative. assume W = 6. Then Z = 1, Y = 5 (2) and X = 12 (1) not valid

Assume W = 5, Then Z = 0, Y = 4, ans X = 9. All valid numbers

The answer 5940

Answer 4

Use the second and third lines to replace W and Z in the first line. This gives you X = W + (W - 1) + (W - 5) = 3W - 6. Now you simply have to try possible values of W. W < 5 would make Z negative and Z > 5 would make X > 10 so W = 5 is needed. You can work out the others from this.

Answer 5

this is multiple - super substitutions

we will use the equation with all four integers

x = w + y + z....and substitute for W
x = (y+1) + y + z
now looking at e3 substitute
z = (Y+1) - 5 = y - 4.....substitute

x = (y+1) + y + (y-4) = now we can solve it

x = 3y - 3

so if y = 4 (just a guess here as >5 results in a 2 digit number)
x = 9
w = 5
z = 0

e 1 is true:therefore: wxyz IS 5940

TADA !

Answer 6

X=W+Y+Z
W=Y+1
Z=W-5
Thus, X = 2Y+1+Z
Z = Y - 4
X = 3Y-3
W=Y+1
So we find legitimate values for Y. Z cannot be negative, so Y is at least 4. X is less than 10 so 10>3Y-3
13>3Y
4 1/3 >Y; Y is at least 4
Thus Y = 4, Z = 0, X = 9, and W = 5
So the integer is 5940.
Please check my answer.

Answer 7

X=W+W-1+W-5
X=3W-6
W≥2
andW ≤5
Z=W-5
W≥5
as W,X,Y,Z cannot be negative or two digit integers
then from the above two equations we deduce that
W=5
then
X=9
Z=0
Y=4
No: 5940

Answer 8

Kapil explained it really well.