Question 1179993
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<pre>

Your matrix

        X1	X2	X3	b
1	2	3	-4	-4
2	1	-2	2	3
3	1	3	-2	1


Write down the main matrix and find its determinant

        X1	X2	X3
1	2	3	-4
2	1	-2	2
3	1	3	-2

Δ = -12


Replace the 1st column of the main matrix with the solution vector and find its determinant

        X1	X2	X3
1	-4	3	-4
2	3	-2	2
3	1	3	-2

Δ1 = -12


Replace the 2nd column of the main matrix with the solution vector and find its determinant

       X1	X2	X3
1	2	-4	-4
2	1	3	2
3	1	1	-2

Δ2 = -24


Replace the 3rd column of the main matrix with the solution vector and find its determinant

        X1	X2	X3
1	2	3	-4
2	1	-2	3
3	1	3	1

Δ3 = -36


x1 = Δ1 / Δ = (-12) / (-12) = 1
x2 = Δ2 / Δ = (-24) / (-12) = 2
x3 = Δ3 / Δ = (-36) / (-12) = 3


Solution set:

x1 = 1
x2 = 2
x3 = 3
</pre>

Solved.


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On Cramer's rule for solving systems of 3 equations in 3 unknowns see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Co-factoring-a-3x3-determinant.lesson>Co-factoring the determinant of a 3x3 matrix</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/HOW-TO-solve-system-of-linear-eqns-in-three-unknowns-using-det.lesson>HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-three-unknowns-using-determinant.lesson>Solving systems of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-by-reducing-them-to-systems-of-linear-equations-in-three-unknowns.lesson>Solving word problems by reducing to systems of linear equations in three unknowns</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
&nbsp;&nbsp;&nbsp;&nbsp; "<U>3x3-Matrices, determinants, Cramer's rule for systems in three unknowns</U>" 



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

<A HREF=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</A>


https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver