4x-4y+4z=8
9x+3y+z=6
16x+4y+z=2

9514 1404 393

Answer:

  (x, y, z) = (-1, 3, 6)

Step-by-step explanation:

The augmented matrix for the system is ...

  [tex]\left[\begin{array}{ccc|c}4&-4&4&8\\9&3&1&6\\16&4&1&2\end{array}\right][/tex]

Your graphing or scientific calculator can tell you the solution to this system is ...

  (x, y, z) = (-1, 3, 6)

__

If you want to solve this by hand, it can work well to divide the first equation by 4 to get ...

  x -y +z = 2

This can be subtracted from the other two equations to eliminate z.

  (9x +3y +z) -(x -y +z) = (6) -(2)  ⇒  8x +4y = 4

  (16x +4y +z) -(x -y +z) = (2) -(2)  ⇒  15x +5y = 0

These two equations can be reduced to standard form:

Subtracting the first equation from the second, we have ...

  (3x +y) -(2x +y) = (0) -(1)  ⇒  x = -1

Substituting into the first gives y:

  2(-1) +y = 1

  y = 3 . . . . . . . add 2

Then we can find z from the reduced first equation above:

  z = 2 -x +y = 2 -(-1) +3 = 6

Then the solution is (x, y, z) = (-1, 3, 6).

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