SOLUTION: given the system of equations 2x-3y-9z=20 x+3z=-2 -3x+y-4z=-2 find the complete solution write x and y as functions of z

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Question 1207192: given the system of equations 2x-3y-9z=20 x+3z=-2 -3x+y-4z=-2 find the complete solution write x and y as functions of z
Found 3 solutions by MathLover1, greenestamps, math_tutor2020:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

....eq.1
...........eg.2
.....eq.3
-----------------------------
start with
....eq.1, solve for

....eq.1a
then
...........eg.2, solve for
...........eg.2a

from eq.1a and eq.2a we have
....solve for
...both sides multiply by

..both sides divide by
.....eq.3a

solutions of and in terms of are:



check:
....eq.1, substitute and



=> true

...........eg.2, substitute

=> true

.....eq.3, substitute and



=>true


Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


[1]
[2]
[3]

The instructions are to write x and y as functions of z. We can get x as a function of z directly from [2]:

[4]

Substitute [4] in [1] and solve to find y as a function of z:





[5]

ANSWER:
z = z
x = -3z-2
y = -5z-8
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!


is equivalent to


That system converts to this augmented matrix.
2-3-920
103-2
-31-4-2

Normally the grid lines aren't present to separate each item. But I decided to make it into a table format.

Let's apply Gauss Jordan Elimination to get the matrix into Reduced Row Echelon Form (RREF).
2-3-920
103-2
-31-4-2

103-2R1 <--> R2
2-3-920
-31-4-2

103-2
0-3-1524R2 - 2*R1 --> R2
-31-4-2

103-2
0-3-1524
015-8R3 + 3*R1 --> R3

103-2
015-8R2 <--> R3
0-3-1524

103-2
015-8
0000R3 + 3*R2 --> R3



Here is a step-by-step calculator that is very useful to row reduce matrices
http://www.math.odu.edu/~bogacki/lat/
It is called "linear algebra toolkit".
Click the "Enter" link and then go to "Row operation calculator". Let me know if you have any questions about this calculator.


More practice with gauss-jordan elimination
https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.1203611.html

--------------------------------------------------------------------------

To briefly summarize, we have gone from this matrix

to this matrix

The row of all zeros tells us that we will have infinitely many solutions. This system is consistent and dependent.

The 2nd matrix converts back to this system

and this is what results when we get each z term to the other side


Therefore each of the infinitely many solutions are of the form (x,y,z) = (-3z-2,-5z-8,z) where z is any real number.

Examples:
If z = 0 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*0-2,-5*0-8,0) = (-2,-8,0) is a solution
If z = 1 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*1-2,-5*1-8,1) = (-5,-13,1) is a solution
If z = 2 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*2-2,-5*2-8,2) = (-8,-18,2) is a solution
If z = 3 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*3-2,-5*3-8,3) = (-11,-23,3) is a solution
All of these solution points are located on the same straight line.

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