.
Consider the circuit in the figure. The currents I1, I2, and I3 in amperes are given by the solution of the system of linear equations.
4I1 + 8I3 = 18
2I2 + 8I3 = 54
I1 + I2 − I3 = 0
Use Cramer's Rule to find the three currents.
I1 =
I2 =
I3 =
~~~~~~~~~~~~~~~~~~~
Your matrix
X1 X2 X3 b
1 4 0 8 18
2 0 2 8 54
3 1 1 -1 0
Write down the main matrix and find its determinant
X1 X2 X3
1 4 0 8
2 0 2 8
3 1 1 -1
Δ = -56
Replace the 1st column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 18 0 8
2 54 2 8
3 0 1 -1
Δ1 = 252
Replace the 2nd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 4 18 8
2 0 54 8
3 1 0 -1
Δ2 = -504
Replace the 3rd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 4 0 18
2 0 2 54
3 1 1 0
Δ3 = -252
x1 = Δ1 / Δ = 252 / (-56) = -9/2
x2 = Δ2 / Δ = (-504) / (-56) = 9
x3 = Δ3 / Δ = (-252) / (-56) = 9/2
Solution set:
x1 = -9/2
x2 = 9
x3 = 9/2
Solved.
---------------
On Cramer's rule for solving systems of 3 equations in 3 unknowns see the lessons
- Determinant of a 3x3 matrix
- Co-factoring the determinant of a 3x3 matrix
- HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)
- Solving systems of linear equations in three unknowns using determinant (Cramer's rule)
- Solving word problems by reducing to systems of linear equations in three unknowns
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns"
In addition, there are many free of charge SOLVERS on a Cramer's rule in the internet.
One of such popular solvers is in this site under the link
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver