Solving Linear Equations in Three Variables

Question

Solve the system:

 

 2x-3y-z=5

-2x+3y+z=-5

8x-12y-4z+20

 

How I understand it is the 2x+-2x along with the -3y+3y and -z+z can cancel each other out.  I am having trouble figuring out what to do next.  Any help will be appreciated!

Answer 1

Hey Courtnee -- its often helpful add/subtract expressions, sometimes doubling/halving them ...

 

here we don't have 3 different expressions: (2) is the negative of (1), (3) is (1) rescaled by 4 :)

Answer 2

We have one equation with three variables:

 

2x-3y-z=5

 

(Your other two equations are just multiples of this equation.)

 

You can solve it for any one variable, eg.

 

z=2x-3y-5

 

which has infinitely many solutions, namely all points of the form

 

(x,y,2x-3y-5), where x and y are any real numbers.

 

Geometrically, the solution set corresponds to a plane in 3-space.