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

Algebra ->  Linear-equations -> 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       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!

2x-3y-9z=20 ....eq.1
x%2B3z=-2 ...........eg.2
-3x%2By-4z=-2.....eq.3
-----------------------------
start with
2x-3y-9z=20 ....eq.1, solve for x
2x=20%2B3y%2B9z+
x=10%2B3y%2F2%2B9z%2F2 ....eq.1a
then
x%2B3z=-2 ...........eg.2, solve for+x
x=-2-3z ...........eg.2a

from eq.1a and eq.2a we have
10%2B3y%2F2%2B9z%2F2=-2-3z....solve for+y
3y%2F2=-2-3z-10-9z%2F2...both sides multiply by 2
3y=-4-6z-20-9z
3y=-15z-24..both sides divide by 3
y=-5z-8.....eq.3a

solutions of x+and y in terms of z+are:
x=-2-3z
y=-5z-8

check:
2x-3y-9z=20 ....eq.1, substitute x and+y
2%28-2-3z%29-3%28-5z-8%29-9z=20
-4-6z%2B15z%2B24-9z=20
-15z%2B15z%2B20=20
20=20=> true

x%2B3z=-2 ...........eg.2, substitute x
-2-3z%2B3z=-2
-2=-2 => true

-3x%2By-4z=-2.....eq.3, substitute x and+y
-3%28-2-3z%29%2B%28-5z-8%29-4z=-2
6%2B9z-5z-8-4z=-2
9z-9z-2=-2
-2=-2 =>true


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


2x-3y-9z=20 [1]
x%2B3z=-2 [2]
-3x%2By-4z=-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]:

x=-3z-2 [4]

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

2%28-3z-2%29-3y-9z=20
-6z-4-3y-9z=20
-3y-15z=24
y%2B5z=-8
y=-5z-8 [5]

ANSWER:
z = z
x = -3z-2
y = -5z-8

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

system%282x-3y-9z=20%2Cx%2B3z=-2%2C+-3x%2By-4z=-2%29
is equivalent to
system%282x-3y-9z=20%2C1x%2B0y%2B3z=-2%2C+-3x%2B1y-4z=-2%29

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
%28matrix%283%2C4%2C2%2C-3%2C-9%2C20%2C1%2C0%2C3%2C-2%2C-3%2C1%2C-4%2C-2%29%29
to this matrix
%28matrix%283%2C4%2C1%2C0%2C3%2C-2%2C0%2C1%2C5%2C-8%2C0%2C0%2C0%2C0%29%29
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
system%28x%2B3z=-2%2Cy%2B5z=-8%29
and this is what results when we get each z term to the other side
system%28x=-3z-2%2Cy=-5z-8%29

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.