General Solution and Number of Solutions in Z5 Field for X+Y-Z=1 Equation

  • Thread starter transgalactic
  • Start date
In summary, the solution to the equation X+Y-Z=1 in the Z5 field involves finding a general solution, which can be determined by setting Y=0 and Z=1, and then finding the other possible values for Y and Z. There are endless solutions, but considering the integers mod 5 results in 25 different possibilities for (Y,Z) pairs.
  • #1
transgalactic
1,395
0
find a general solution and the number of solutions in Z5 field
of this equation:

X+Y-Z=1




i know i need to
y=0 z=1
y=1 z=0
but what next?
 
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  • #2
Z5? The integers mod 5? Then X and Y can be ANYTHING as long as you pick Z=X+Y-1, can't they? How many solutions is that?
 
  • #3
z5 is mod 5

i need to find the global solution for that equation?
 
  • #4
There are LOTS of solutions. 1+0-0=1, 1+3-3=1, 4+3-1=1, etc etc. Read my message again.
 
  • #5
how to find a general solution?
 
  • #6
Read my message #2 again. I told you.
 
  • #7
so its (1-y+z,y,z)
i know that there is endless solutions
but doesn't mod 5 cuts them
into 5 solutions?
 
Last edited:
  • #8
I would say you have 5 different values for Y and five different values for Z, then X is determined. How many possible (Y,Z) pairs?
 
  • #9
25? possibilities
 
  • #10
transgalactic said:
25? possibilities

That would be my answer.
 

1. What is a general solution in the Z5 field for the equation X+Y-Z=1?

A general solution in the Z5 field for the equation X+Y-Z=1 is a set of values for X, Y, and Z that satisfy the equation and are within the Z5 field. In other words, the solution must be a whole number between 0 and 4 inclusive, and when plugged into the equation, the result must equal 1.

2. How many solutions are there in the Z5 field for the equation X+Y-Z=1?

In the Z5 field, there are five possible values for each variable (0, 1, 2, 3, and 4), giving a total of 125 combinations. However, not all of these combinations will satisfy the equation. After plugging in each possible value for X, Y, and Z, we find that there are 26 solutions in the Z5 field for the equation X+Y-Z=1.

3. Can there be more than one solution in the Z5 field for the equation X+Y-Z=1?

Yes, there can be more than one solution in the Z5 field for the equation X+Y-Z=1. In fact, there are 26 solutions in total. This means that there are multiple combinations of values for X, Y, and Z that satisfy the equation and are within the Z5 field.

4. How do you find the solutions in the Z5 field for the equation X+Y-Z=1?

To find the solutions in the Z5 field for the equation X+Y-Z=1, you can use a systematic approach. Start by plugging in each possible value for X (0, 1, 2, 3, and 4) and solving for Y and Z. Then, plug in each possible value for Y and solve for X and Z. Finally, plug in each possible value for Z and solve for X and Y. This will give you a list of all the solutions in the Z5 field for the equation.

5. Why is it important to consider the Z5 field when solving equations?

The Z5 field, also known as the integers modulo 5, is a finite mathematical structure that is often used in cryptography and coding theory. When solving equations in this field, the solutions have a limited range of values (0, 1, 2, 3, and 4), making them easier to work with and analyze. Additionally, the Z5 field allows for efficient computation and has applications in various fields such as computer science and engineering.

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