SOLUTION: Solve the equation 2x−5y+9z=16. [x, y, z] = [_, _, _]+s[_,_,_]+t[_,_,_] FILL IN THE BLANKS

Algebra ->  Matrices-and-determiminant -> SOLUTION: Solve the equation 2x−5y+9z=16. [x, y, z] = [_, _, _]+s[_,_,_]+t[_,_,_] FILL IN THE BLANKS      Log On


   

Question 1159569: Solve the equation
2x−5y+9z=16.
[x, y, z] = [_, _, _]+s[_,_,_]+t[_,_,_]
FILL IN THE BLANKS
Answer by ikleyn(53979) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the equation
2x−5y+9z=16.
[x, y, z] = [_, _, _] + s[_,_,_] + t[_,_,_]
FILL IN THE BLANKS
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To help you better understand the problem, I will re-formulate it in normal human language. 

You are given an equation

    2x - 5y + 9z = 16.    (1)


They want you to represent the general solution as the sum of a particular solution
and the general solution of the associated homogeneous linear equation

    2x - 5y + 9z = 0.    (2)


The implementation is as follows. From equation (1), you have

    x = 8 + 2.5y - 4.5z    (3)

as the general solution (y and z are arbitrary/free/independent variables, x is the dependent variable).


So, a particular solution is x = 8, y = 0, z = 0.


Next, we introduce parameters s = y and z = t.

Then we can write the general solution (3) in the form


    [x,y,z] = [8,0,0] + s[0,1,0] + t[0,0,1].    (4)


This form (4) is exactly what the problem does request.

Solved.