SOLUTION: Consider the following system of equations. 4x + ay = −1 5x + by − 5z = −2 5x + cy + 3z = 0 And let A = 4 a 0 5 b −5 5 c 3 Suppose

Algebra ->  Matrices-and-determiminant -> SOLUTION: Consider the following system of equations. 4x + ay = −1 5x + by − 5z = −2 5x + cy + 3z = 0 And let A = 4 a 0 5 b −5 5 c 3 Suppose      Log On


   

Question 1084343: Consider the following system of equations.
4x + ay = −1
5x + by − 5z = −2
5x + cy + 3z = 0
And let
A =
4 a 0
5 b −5
5 c 3

Suppose that det(A) = 5. Use Cramer's rule to solve the above system for the variable y
Answer by ikleyn(52767) About Me  (Show Source):
You can put this solution on YOUR website!
.
The main/major coefficient matrix is

A = %28matrix%283%2C3%2C+4%2Ca%2C0%2C+5%2Cb%2C-5%2C+5%2Cc%2C3%29%29.


Its determinant is D = det (A) = 5, according to the condition.


To find the unknown "y", you must, according to the Cramer's rule, replace the second column by the vector of the right side terms,

and then calculate the determinant Dy of this modified matrix.



    So, go to 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



You will find there free of charge online solver to calculate the determinant of a 3x3-matrix.


Input the modified matrix there and press the "Solve" button.

You will get the determinant Dy. It is 16.


So, your solution for "y" is y = %28Dy%29%2FD = 16%2F5, according to the Cramer's rule.


The solution is completed.


    So, using this methodology, you are relieved of manual calculating determinants.
    But you still must know how the Cramer's rule works.

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