How do you solve a system of equations?

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?

Answer 1

So the 4 digit number is WXYZ and I guess you're suppose to determine the integer.

And the digits W, X, Y, and Z are related as follows:

 

X = W + Y + Z  ........ eq 1

W = Y + 1         ............eq 2

Z  = W - 5         ............ eq 3

 

Please keep in mind that you're working with nonnegative integers which are strictly less than 10 (or digits)

Some immediate observations:  Be sure you understand how they were attained!!!

 

 1.  0 <= X <= 9    (because 0<= W + Y + Z <= 9)

 2.  W + Y >= 9     (because W >= 5 and Y >= 4 implies W + Y >= 5 + 4 = 9)

 3.  W >= 5           (because Z is a digit and so Z >=0  or W - 5 >=0  implies W >=5)

 4.  Y >= 4            (because W >= 5  implies that Y + 1 >=5 or Y >=4 )

 5.  Z = 0              (because X = W + Y + Z and W + Y >= 9 implies that Z = 0)

 

From the above observations we now have the following important results.

                               W = 5,  Y = 4   X = 9 and Z = 0  and W + Y = 9

 

Now let's form the number or positive integer WXYZ which is 5940.

 

 

Answer 2

second equation: w = y + 1

third equation: w = z + 5

subtract third equation from second equation

 

w - w = y + 1 - z - 5

0 = y - z - 4

4 = y - z

 

If z = 1, then y = 5, then w = 6, and x = 12 which is impossible.

Therefore Z must be less than 1 which makes it zero.

Then y = 4, w = 5, and x = 9

 

WXYZ = 5940

Answer 3

Normally you need as many different equations as unknown variables to solve a system of equations.

This problem has some extra constraints that help, namely the unknowns are different digits and there are only 10 digits to choose. Also, since  W + Y + Z  is a digit, the sum is less than 10. If W-5 is digit Z, then W ≥ 5. If you let W = 5, then Z = 0 and Y = 4, leaving X = 9.

Checking shows 9 = 5+4+0; 5 = 4+1; and 0 = 5-5.

So, the number is 5940.