Why Are 3x3 Systems with Arithmetically Increasing Constants Always Infinite?

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In summary, the conversation discusses a system of equations where the solution is always infinite. The constants in each equation increase or decrease by a constant amount, leading to a pattern in the augmented matrix. By using column operations, it is shown that the last unknown value, z, can be any value and still satisfy the equations. However, there is no clear explanation as to why this pattern occurs and further help is requested in understanding it.
  • #1
AlphaNoodle
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I encountered a system:
3x+5y+7z = 9
7x+3y-z=-5
12x+13y+14z=15
And the solution was infinite solutions.

However, when looking at each equation, the constants (including coefficients) increase/decrease by a constant amount.
3x+5y+7z = 9 (+2)
7x+3y-z=-5 (-4)
12x+13y+14z=15 (+1)

And I made other systems using that same format, where all the equations' constants increased/decreased arithmetically (by a constant)
And the solution was always infinity.

I am curious to why this is and is there any proof behind this?
I got as far as:

Ax+(A+m)y+(A+2m)z=(A+3m)
Bx+(B-n)y+(B-2n)z=(B-3n)
Cx+(C+k)y+(C+2k)z=(C+3k)

But I am clueless to why there is always infinite solutions to these types of systems. I always get to the point where a number = a number (using elimination) to solve the system, and form there the solutions are always infinite. I don't even know how to proceed with a proof, and I was hoping for some help. I am sure I am interpreting something wrong or missing something, but anything would be appreciated. Thanks in advanced.
 
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  • #2
One way to solve a system of equations is to "column" reduce the augmented matrix.

Here, the augmented matrix is
[tex]\begin{bmatrix}A & A+ n & A+ 2n & A+ 3n \\ B & B- m & B- 2m & B- 3m \\ C & C+ k & C+ 2k & C+ 3k\end{bmatrix}[/tex]

If we add -1 times the first column to each of the other columns we get
[tex]\begin{bmatrix}A & n & 2n & 3n \\ B & -m & -2m & -3m \\ C & k & 2k & 3k\end{bmatrix}[/tex]

Now add -2 times the second column to the third column and -3 times the second column to the fourth column to get
[tex]\begin{bmatrix}A & n & 0 & 0 \\ B & -m & 0 & 0 \\ C & k & 0 & 0\end{bmatrix}[/tex]
and now we can see that, since the last two columns are all "0"s, the last unknown value, z, can be anything at all and still satisfy these equations.
 
  • #3
I did not know there were column operations, still in high school haha. But a question I have is how do you represent column operations? in other words row operations are simply L1 -> L1+3 or something like that, but how are column operations represented?
 

FAQ: Why Are 3x3 Systems with Arithmetically Increasing Constants Always Infinite?

Why are 3x3 systems with arithmetically increasing constants always infinite?

The reason for this is because when the constants in a 3x3 system are arithmetically increasing, it means that each constant is larger than the one before it. This creates a pattern where the equations in the system will always have a solution, no matter what numbers are used for the variables. This results in an infinite number of solutions.

Can a 3x3 system with arithmetically increasing constants have a finite number of solutions?

No, a 3x3 system with arithmetically increasing constants will always have an infinite number of solutions. This is because the pattern of increasing constants ensures that there will always be a solution for the equations in the system, no matter what numbers are used for the variables.

How does the property of arithmetically increasing constants affect the solutions of a 3x3 system?

The property of arithmetically increasing constants guarantees that there will always be an infinite number of solutions for a 3x3 system. This is because the increasing constants create a pattern where the equations in the system will always have a solution, regardless of the values used for the variables.

Are there any other types of systems that will always have an infinite number of solutions?

Yes, there are other types of systems that will always have an infinite number of solutions. For example, a 3x3 system with arithmetically decreasing constants will also have an infinite number of solutions, as well as a 3x3 system with geometrically increasing or decreasing constants.

How does the concept of infinity relate to 3x3 systems with arithmetically increasing constants?

The concept of infinity is closely related to 3x3 systems with arithmetically increasing constants because the increasing constants create a pattern that results in an infinite number of solutions. This concept highlights the idea that there is no limit to the number of solutions for these types of systems, as they will always have an infinite number of solutions.

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