SOLUTION: Need help solving system with three variables -2x + 8y + 2z = 4 x + 6y + 3z = 4 3x - 2y + z = 0

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Question 1069422: Need help solving system with three variables
-2x + 8y + 2z = 4
x + 6y + 3z = 4
3x - 2y + z = 0
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Need help solving system in three variables
-2x + 8y + 2z = 4
x + 6y + 3z = 4
3x - 2y + z = 0
~~~~~~~~~~~~~~~~~~~ 

-2x + 8y + 2z = 4    (1)
  x + 6y + 3z = 4    (2)
 3x - 2y +  z = 0    (3)


Add equations (1) and (3) (both sides). You will get

  x + 6y + 3z = 4    (4).

Now compare equation (4) with equation (2). What do you see ??

Their left sides are identical. Their right sides are identical, too.


What does it mean ??  - It means that the equation (2) is DEPENDENT on equations (1) and (3).
In other words, equation (2) doesn't carry new information.


What does it mean ??  - It means that, factually, there are only TWO equations for THREE unknowns.


What does it mean ??  - It means that the original system of equations HAS INFINITELY MANY solutions.


Answer. The system of equations (1), (2), (3) HAS INFINITELY MANY solutions.

There is a bunch of lessons on solving systems of linear equations in three unknowns by the Substitution and Elimination methods
    - Solving systems of linear equations in 3 unknowns by the Substitution method
    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Substitution method
    - Solving systems of linear equations in 3 unknowns by the Elimination method
    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Elimination method
    - OVERVIEW of LESSONS on solving systems of linear equations in three unknowns by the Substitution and/or Elimination methods

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"


Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Need help solving system with three variables
-2x + 8y + 2z = 4
x + 6y + 3z = 4
3x - 2y + z = 0


x + 6y + 3z = 4 ------- Adding eqs (i) & (iii) ------- eq (iv)

If you inspect eqs (iv) & (ii), you'll see that they are IDENTICAL. What do you think this means? If you don't know, then you need to look it up. 

I think I've done the harder part of the work for you. 

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