Solving Systems of Six Equations with Nine Variables

In summary, the conversation discusses Archetype J, a system with six equations and nine variables that is consistent and has a null space of coefficient matrix with dimension 5. The speaker mentions that there are several sample solutions and verifies that one of them is correct. They then explain that due to the number of variables being greater than the number of equations, the system has infinitely many solutions. The speaker also mentions not understanding the meaning of "Null space of coefficient matrix has dimension 5."In response to the speaker's examples of two lines with different equations, the conversation continues to discuss the number of solutions for each and the determinant of their coefficient matrices.
  • #1
karush
Gold Member
MHB
3,269
5
The details for Archetype J (System with
six equations,
nine variables.
Consistent.
Null space of coefficient matrix has dimension 5.)
include several sample solutions.

Verify that one of these solutions is correct (any one, but just one).
Based only on this evidence, and especially without doing any row operations,
explain how you know this system of linear equations has infinitely many solutions

Ok the only thing I can think of the there is more variables than equations so you cannot have a unique solution
also I didn't know exactly what "Null space of coefficient matrix has dimension 5" meant
 
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  • #2
1) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = 3x - 7\). How many solutions does this have?

2) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = -ax - b\). How many solutions does this have?

3) Say you have two lines \(\displaystyle y = ax + b\) and \(\displaystyle y = ax\). (\(\displaystyle b \neq 0\).) How many solutions does this have?

Now take a look at the coefficient matrix for each. What is the determinant of each? What does that tell you?

-Dan
 

1. How do you solve a system of six equations with nine variables?

Solving a system of six equations with nine variables requires using a method called elimination. This involves manipulating the equations to eliminate one variable at a time until you are left with a system of three equations and three variables, which can then be solved using substitution or another method.

2. Is it possible to have more than one solution when solving a system of six equations with nine variables?

Yes, it is possible to have more than one solution when solving a system of six equations with nine variables. This is known as a system of equations with infinite solutions. In this case, the equations are dependent on each other and can be satisfied by multiple values for the variables.

3. What is the importance of solving systems of six equations with nine variables?

Solving systems of six equations with nine variables is important in many fields of science and mathematics, such as engineering, physics, and economics. It allows us to find the values of multiple variables that satisfy a set of equations, which can then be used in real-world applications.

4. Can technology be used to solve systems of six equations with nine variables?

Yes, technology can be used to solve systems of six equations with nine variables. There are many online calculators and software programs that can handle complex systems of equations and provide solutions. However, it is still important to understand the underlying concepts and methods for solving these systems.

5. Are there any shortcuts or tricks for solving systems of six equations with nine variables?

There are no specific shortcuts or tricks for solving systems of six equations with nine variables, as each system may require a different approach. However, practicing and understanding the methods of elimination and substitution can make the process more efficient. Additionally, it can be helpful to first simplify the equations by combining like terms or using the distributive property before attempting to solve the system.

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