# General Solution to System of Equations w/o Eigenvalues

• hbomb
In summary, the conversation discusses solving a system of equations using eigenvalues and eigenvectors, but also mentions an easier method using a lower triangular matrix. The participants also suggest asking the professor for clarification on the topic.
hbomb
I have a question that involves a system of equations that I can't figure out

Give the general solution of the set of equations below:

x'=2x
y'=-x+3y
z'=2x-4y+6z

Hint: While you can use eigenvalues and eigenvectors for this one, there is an easier way to do it.

That's where I'm stuck, I know how to do this using the eigenvalues and eigenvectors.

This is a lower triangular matrix, so the eigenvalues are 2, 3, 6. But what's the other way of doing this?

If x'=2x, what does x equal?

Plug that into the x for y' = -x + 3y, and solve for y.

Do the same for z then.

I find it a bit disconcerting that you have posted so many problems about these types of things, you may want to ask your professor to go over exactly what you're doing, since it looks like you just memorized the matrix formula and left it at that

Yea, he's very vague on some of these topics. He gives examples of finding determinants of matrices and finding eigenvalues and eigenvectors, but he never showed how to solve a system of equations with a given point.

## What is a general solution to a system of equations without eigenvalues?

A general solution to a system of equations without eigenvalues is a solution that satisfies all equations in the system without using eigenvalues. It can be found using various methods such as substitution, elimination, or the Gauss-Jordan method.

## Why is finding a general solution important in mathematics and science?

Finding a general solution is important because it allows us to find all possible solutions to a system of equations. This can be useful in solving real-world problems and understanding mathematical and scientific concepts.

## What are some common methods for finding a general solution to a system of equations without eigenvalues?

Some common methods for finding a general solution include substitution, elimination, and the Gauss-Jordan method. These methods involve manipulating the equations to eliminate variables and solve for the remaining ones.

## Can a system of equations have multiple general solutions?

Yes, a system of equations can have multiple general solutions. This means that there can be more than one set of values that satisfy all equations in the system. These solutions may differ depending on the method used to find them.

## How can understanding general solutions help in solving more complex problems?

Understanding general solutions can help in solving more complex problems by providing a foundation and understanding of solving systems of equations. Once the concept of general solutions is understood, it can be applied to more complex systems of equations and problems in various fields of mathematics and science.

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