Solving Systems of Equations with Whole Numbers

Thus, the resulting $x$ will consist of whole numbers as well.In summary, for both given systems (1) and (2), if the determinants of $A$ are -1 and 2 respectively, and all members of $A$ and $b$ are whole numbers, then the solution $x$ will also consist of whole numbers. This can be achieved by finding $A^{-1}$ using the adjugate method, which preserves the whole number values of the original matrix.
  • #1
Yankel
395
0
Hello all

I have a couple of short questions, both similar, which I do not know how to even start, and I could use some help.

1) Show that if in the system Ax=b, det(A)=-1, and all the members of A are whole numbers (belong to Z), and all the members of b are whole numbers (belong to Z), then a single solution exists, and it's members are also whole numbers.

2) Show that if in the system Ax=b, det(A)=2, and all the members of A are whole, and all the members of b are even, then a single solution exists, and all it's members are whole numbers.

Thank you in advance !
 
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  • #2
The solution is $x=A^{-1}b$. One way to find $A^{-1}$ is $\frac{1}{\det A}\text{adj}\,A$ where $\text{adj}\,A$ is the adjugate of $A$. By construction, if $A$ consists of whole numbers, then so does $\text{adj}\,A$. In the second case, one can factor out 2 from $b$ and cancel it with $\det A$.
 

FAQ: Solving Systems of Equations with Whole Numbers

What is a system of equations?

A system of equations is a set of equations that are related and need to be solved together. Typically, there are multiple unknown variables that need to be determined in order to find a solution.

What are whole numbers?

Whole numbers are numbers that are positive, non-decimal, and non-fractional. They include all natural numbers (1, 2, 3, etc.) and the number 0.

How do you solve a system of equations with whole numbers?

To solve a system of equations with whole numbers, you can use various methods such as substitution, elimination, or graphing. The goal is to find the values of the unknown variables that make all of the equations in the system true.

Why is it important to solve systems of equations with whole numbers?

Solving systems of equations with whole numbers is important because it allows us to find the values of unknown variables in a given situation. This can help us make predictions, analyze relationships, and solve real-world problems.

What are some real-life applications of solving systems of equations with whole numbers?

Solving systems of equations with whole numbers has many practical applications, such as in engineering, economics, and science. For example, it can be used to determine the optimal production levels for a business, calculate the trajectory of a rocket, or find the most efficient route for a delivery truck.

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