Given the following system equations:
-Put them into Standard Form, and
-Simplify using Cramer’s rule
IS + G1 (VA + VB) – G2VB = 0
G2VB + G1 (VA – VB) = 0
To put the given system of equations into standard form, let's rearrange them:
1) IS + G1 (VA + VB) – G2VB = 0
2) G2VB + G1 (VA – VB) = 0
Now, let's simplify the system using Cramer's rule. Cramer's rule is a method used to solve systems of linear equations by finding the determinants of matrices.
Step 1: Write the equations in matrix form.
We can represent the given equations in matrix form as:
| 1 G1 -G2 | | IS | | 0 |
| G1 -G1 G2 | × | VA | = | 0 |
| 0 G2 0 | | VB | | 0 |
Step 2: Find the determinant of the coefficient matrix.
The determinant of the coefficient matrix is given by:
D = | 1 G1 -G2 |
| G1 -G1 G2 |
| 0 G2 0 |
D = (1 × (-G1 × 0)) - (G1 × (-G2 × 0)) = 0
Step 3: Find the determinants for each variable.
We now need to find the determinants D_IS, D_VA, and D_VB.
D_IS: We replace the first column of the coefficient matrix with the column containing the constants.
D_IS = | 0 G1 -G2 |
| 0 -G1 G2 |
| 0 G2 0 |
D_IS = (0 × (-G1 × 0)) - (0 × (-G2 × 0)) = 0
D_VA: We replace the second column of the coefficient matrix with the column containing the constants.
D_VA = | 1 0 -G2 |
| G1 0 G2 |
| 0 G2 0 |
D_VA = (1 × (0 × 0)) - (0 × (G2 × 0)) = 0
D_VB: We replace the third column of the coefficient matrix with the column containing the constants.
D_VB = | 1 G1 0 |
| G1 -G1 0 |
| 0 G2 0 |
D_VB = (1 × (-G1 × 0)) - (G1 × (0 × 0)) = 0
Step 4: Solve for the variables.
Using Cramer's rule, we can determine the solution for the system of equations.
IS = D_IS / D = 0 / 0 (indeterminate)
VA = D_VA / D = 0 / 0 (indeterminate)
VB = D_VB / D = 0 / 0 (indeterminate)
Since the determinants D_IS, D_VA, and D_VB are all 0, the system of equations has an indeterminate solution. This means that there are infinitely many solutions or no solutions to the system.