The given system consists of equations of planes. Let's use the Elimination Method to find a solution to this system. Notice that in the third equation there is no

c -

term.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 4 a + 2 b + c = 2 5 a - 3 b + 2 c = 17 a - 5 b = 3 (I) (II) (III)

Currently, none of the terms in this system will cancel out. However, if we multiply (I) by

- 2

the coefficient of

c

in this equation will be the additive inverse of the coefficient of

c

in the second equation; they will add to be

0 .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 2 (4 a + 2 b + c) = - 2 (2) 5 a - 3 b + 2 c = 17 a - 5 b = 3 ⇓ ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 8 a - 4 b - 2 c = - 4 5 a - 3 b + 2 c = 17 a - 5 b = 3

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

c - terms .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 8 a - 4 b - 2 c = - 4 5 a - 3 b + 2 c = 17 a - 5 b = 3 (I) (II) (III)

SysEqnAdd

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

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 8 a - 4 b - 2 c + (5 a - 3 b + 2 c) = - 4 + (17) 5 a - 3 b + 2 c = 17 a - 5 b = 3

RemovePar

$(I):$ Remove parentheses

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 8 a - 4 b - 2 c + 5 a - 3 b + 2 c = - 4 + 17 5 a - 3 b + 2 c = 17 a - 5 b = 3

AddSubTerms

$(I):$ Add and subtract terms

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 3 a - 7 b = 13 5 a - 3 b + 2 c = 17 a - 5 b = 3

Next, we will use our two equations that are only in terms of

a

and

b

to solve for the value of one of the variables. We will once again apply the Elimination Method, but this time it will be similar to when using it in a system with only two variables.

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