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Graphing Equations in Three Variables

Example

A dependent system of three equations

Solve the system:

(1)

(2)

(3)

2x

-6x

4x

-

+

-

3y

9y

6y

-

+

-

z

3z 

2z

  = 4

  = -12

  = 8

Solution

We will first eliminate x from Eqs. (1) and (2). Multiply Eq. (1) by 3 and add the resulting equation to Eq. (2):

6x

-6x

-

+

9y

9y

-

+

3z

3z

= 12

= -12

Eq. (1) multiplied by 3

Eq. (2)

        0 = 0  

The last statement is an identity. The identity occurred because Eq. (2) is a multiple of Eq. (1). In fact, Eq. (3) is also a multiple of Eq. (1). These equations are dependent. They are all equations for the same plane. The solution set is the set of all points on that plane,

{(x, y, z) | 2x - 3y - z = 4}.

 

Helpful Hint

If you recognize that multiplying Eq. (1) by -3 will produce Eq. (2), and multiplying Eq. (1) by 2 will produce Eq. (3), then you can conclude that all three equations are equivalent and there is no need to add the equations.

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