The given system consists of equations of planes. Notice that the coefficient of

z

in the first equation is the additive inverse of the coefficient of

z

in the second equation; they will add to be

0 .

Let's use the Elimination Method to find a solution to this system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - y + z = - 2 x + 3 y - z = 10 x + 2 z = - 8 (I) (II) (III)

We can start by adding the second equation to the first equation to eliminate the

z - terms .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - y + z = - 2 x + 3 y - z = 10 x + 2 z = - 8 (I) (II) (III)

$(I):$ Add $(II)$

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - y + z + (x + 3 y - z) = - 2 + 10 x + 3 y - z = 10 x + 2 z = - 8

$(I):$ Remove parentheses

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - y + z + x + 3 y - z = - 2 + 10 x + 3 y - z = 10 x + 2 z = - 8

$(I):$ Add terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 x + 2 y = 8 x + 3 y - z = 10 x + 2 z = - 8

Having eliminated the

z -

variable from the first equation, we can continue by creating additive inverse coefficients for

z

in the second and third equations. Then we can add or subtract these equations to eliminate

z

from the second equation.

Exercise 4 Page 171

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