**Solve special systems**

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** **Linear systems with three variables may be inconsistent or may include dependent equations. The next examples illustrate these cases.

**Solving an Inconsistent System with Three Variables**

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Solve the system.

2x – 4y + 6z = 5 (1)

-x + 3y – 2z = -1 (2)

x – 2y + 3z = 1 (3)

eliminate x by adding equations (2) and (3) to get the equation

y + z = 0.

Now , eliminate x again, using equation (1) and (3).

-2x + 4y – 6z = -2 Multiply each side of (3) by -2.

2x – 4y + 6z = 5 (1)

0 = 3 False

The resulting false statement indicates that equations (1) and (3) have no common solution. Thus, the system is inconsistent and the solution set is ø. The graph of this system would show these two planes parallel to one another.