Question 1179319
.
Solve the system with three variables by Cramer's Rule.
-2x -y -3z = 3
2x -3y +z = -13
2x -3z = -11
~~~~~~~~~~~~~~~~~~~


<pre>
Your matrix

        X1	X2	X3	b
1	-2	-1	-3	3
2	2	-3	1	-13
3	2	0	-3	-11


Write down the main matrix and find its determinant

        X1	X2	X3
1	-2	-1	-3
2	2	-3	1
3	2	0	-3

Δ = -44


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

        X1	X2	X3
1	3	-1	-3
2	-13	-3	1
3	-11	0	-3

Δ1 = 176


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

        X1	X2	X3
1	-2	3	-3
2	2	-13	1
3	2	-11	-3

Δ2 = -88


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

        X1	X2	X3
1	-2	-1	3
2	2	-3	-13
3	2	0	-11

Δ3 = -44


x1 = Δ1 / Δ = 176 / (-44) = -4
x2 = Δ2 / Δ = (-88) / (-44) = 2
x3 = Δ3 / Δ = (-44) / (-44) = 1


Solution set:

x1 = -4
x2 = 2
x3 = 1
</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