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Hunter-Best [27]
3 years ago
5

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

Mathematics
1 answer:
son4ous [18]3 years ago
4 0
Using Gaussian elimination:
x + y + z = 9  / * ( - 2 )    / ( - 1 ) ( we will multiply by - 2 and add to the 2nd         2 x - 3 y + 4 z = 7      equation and multiply by - 1 and add to the 3rd equation ) 
x - 4 y + 3 z = - 2
-----------------------------
x + y + z = 9
    - 5 y + 2 z = - 11  / * ( - 1 )
    - 5 y + 2 z = - 11
---------------------------
x + y + z = 9
   - 5 y + 2 z = - 11
              0 z = 0
There are infinitely many solutions to the system of equations.
z = t,
- 5 y + 2 t = - 11
5 y = 11 + 2 t
y = ( 11 + 2 t ) / 2
x + ( 11 + 2 t ) / 2  + t = 9    / * 2
2 x + 11 + 2 t + 2 t = 18
2 x = 18 - 11 - 4 t
x = ( 7 - 4 t ) / 2
( x, y , z ) = (   (7 - 4 t ) / 2,   ( 11 + 2 t ) / 2,    t  )
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arlik [135]

Answer:

  x = 0; y = -6; z = 1

Step-by-step explanation:

You want to find the solution to the system of equations ...

  • 9x+y-3z=-9
  • 10x-y+2z=8
  • -10x-y+4z=10

<h3>Solution</h3>

This set of equations is conveniently solved using the matrix row-reduction features of a scientific or graphing calculator, spreadsheet, or any of a number of apps or on-line calculators.

The attachment shows the solution to be ...

  (x, y, z) = (0, -6, 1)

__

<em>Additional comment</em>

Solving a system of three or more equations "by hand" often can be done by an ad hoc process. It can be done in systematic fashion using Gauss-Jordan reduction techniques, but that often gets messy. Similarly, Cramer's Rule can be used, but that math tends to involve more arithmetic operations than are really necessary.

Adding the first equation to the other two eliminates the y-variable and reduces the system to two equations in two unknowns.

  (9x +y -3z) +(10x -y +2z) = (-9) +(8) . . . . adding [1] and [2]

  19x -z = -1 . . . . simplified

  (9x +y -3z) +(-10x -y +4z) = (-9) +(10) . . . . adding [1] and [3]

  -x +z = 1 . . . . simplified

At this point, we could graph the two equations, or we can proceed algebraically.

Adding these two equations gives ...

  (19x -z) +(-x +z) = (-1) +(1)

  18x = 0   ⇒   x = 0

Using the second of the reduced equations, we find ...

 -x +z = 1

  0 +z = 1   ⇒   z = 1

And using the second of the original equations, gives us ...

  10(0) -y +2(1) = 8

  -y = 6 . . . . . subtract 2

  y = -6 . . . . multiply by -1

Then the solution is (x, y, z) = (0, -6, 1), as above.

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