Question

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

Answer 1

9514 1404 393

Answer:

(x, y, z) = (-1, 3, 6)

Explanation:

The augmented matrix for the system is ...

`\left[\begin{array}c4&-4&4&8\\9&3&1&6\\16&4&1&2\end{array}\right]`

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

(x, y, z) = (-1, 3, 6)

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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:

• 2x +y = 1
• 3x +y = 0

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).