Cramers Rule and unique solutions

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In summary, the conversation discusses finding values for s and t that result in a unique solution for a given system of equations. While s=2 and t=1 do not have a unique solution, there are other possible values such as s=4 and t=2 that do. The conversation also mentions using Cramer's rule to find solutions, relying on the determinant of the coefficient matrix being non-zero.
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nick484
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Hey

Im having trouble of how to go about this. Afterwards we have to perform some Cramers rule operations however I am currently struggling. For:

Find all values of s and t for which the following system has a unique solution:
sx1 + 2x2 + x3 + 3x4 = 1
tx1 + x2 + 3tx3 + s^2x4 = 0
sx3 + 2tx4 = 2
x3 + x4 = 1

Do I just say that (s/2)=/1 and t=/1 because of the last two equations or is there more to it?

Thanks
 
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  • #2
Well, s= 2, t= 1 will cause there to be no solution but those are not the only values. For example, if s= 4 and t= 2, the final two equations wil be 4x3+ 4x4= 2 which reduces to x3+ x4= 1/2, contradicting the last equation.

Since you talk about Cramer's rule, I assume you know that you can write the solutions to such a system as one determinant divided by the other- and you can do that as long as the denominator, the determinant of the coefficient matrix, is non-zero.
 
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Thanks
 

FAQ: Cramers Rule and unique solutions

1. What is Cramer's Rule?

Cramer's Rule is a method used to solve systems of linear equations by using determinants. It allows us to find unique solutions to systems that have the same number of equations and unknown variables.

2. How does Cramer's Rule work?

Cramer's Rule involves finding the determinants of the coefficient matrix and the constant matrix in a system of linear equations. These determinants are then used to calculate the values of each variable in the system, resulting in a unique solution.

3. What are the requirements for using Cramer's Rule?

Cramer's Rule can only be used for systems of linear equations that have the same number of equations and unknown variables. The coefficient matrix must also have a non-zero determinant in order for Cramer's Rule to be applicable.

4. Can Cramer's Rule always find a unique solution?

No, there are cases where Cramer's Rule may not be able to find a unique solution. This can happen if the coefficient matrix has a determinant of zero or if there are infinitely many solutions to the system of equations.

5. Are there any alternatives to using Cramer's Rule?

Yes, there are other methods for solving systems of linear equations such as elimination, substitution, and matrix inversion. These methods may be more efficient or easier to use depending on the specific system of equations.

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