Using cramer's Rule to solve for x2, i'm confused on this example

In summary, The conversation discusses how to use Cramer's rule to find the unknown variables in a system of equations represented in matrix form. The speaker mentions verifying determinants and replacing columns in the matrix to find the solution. They also reference an image for clarification.
  • #1
mr_coffee
1,629
1
I scanned the page out of the book, alittle bit is kinda burly but I'm confused on how they got A_2(B), where we are thinking of the system in matrix form AX = B. One verfies that detA = -127 and det[A_2(B)] = 313, so
x2 = det[A_2(B)]/det(A) = 313/-127 = -313/127;

How did they get B? and det[A_2(B)] here is the picture:
http://img416.imageshack.us/img416/9428/lastscan5eu.jpg
thanks.
 
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  • #2
nevermind i found the pattern, all they do is replace the 2nd column with the values after the =.
 
  • #3
Correct, to find the 'n-th' unknown with Cramer's rule, you have to replace the n-th column by the column of the constants.
 

1. What is Cramer's Rule?

Cramer's Rule is a method used to solve systems of linear equations by using determinants. It involves creating a matrix of coefficients and a matrix of constants, and then using the determinants of these matrices to find the values of the variables.

2. How do I use Cramer's Rule to solve for x2?

To solve for x2 using Cramer's Rule, you will need to have a system of equations with two variables, x1 and x2, and two equations. Then, you will need to create a matrix of coefficients and a matrix of constants, and use these to find the determinants. Finally, divide the determinant of the x2 matrix by the determinant of the coefficient matrix to find the value of x2.

3. Can you provide an example of solving for x2 using Cramer's Rule?

Sure! Let's say we have the following system of equations:
2x1 + 3x2 = 8
4x1 + 5x2 = 12
We can create the following matrices:
Coefficient matrix:
| 2 3 |
| 4 5 |
Constant matrix:
| 8 |
| 12 |
Then, we find the determinants of these matrices:
Coefficient determinant = (2*5) - (3*4) = 2
x2 matrix determinant = (8*5) - (12*3) = 20 - 36 = -16
Finally, we divide the x2 matrix determinant by the coefficient determinant:
-16 / 2 = -8
Therefore, x2 = -8.

4. Is Cramer's Rule the only method for solving systems of linear equations?

No, Cramer's Rule is just one of many methods for solving systems of linear equations. Other common methods include substitution, elimination, and graphing. The method you use may depend on the specific equations and variables involved, as well as personal preference.

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

Yes, there are some limitations to using Cramer's Rule. It can only be used to solve systems of equations with the same number of equations as variables, and it becomes more complex and computationally intensive as the number of variables increases. In addition, if any of the determinants involved are equal to 0, Cramer's Rule cannot be used.

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