Let's start by reviewing how we can solve a system of equations by graphing. When graphing a system of three equations with three variables, each equation represents a plane in a coordinate space. The intersections of these planes are the solution to our system. Let's have a look to our system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - 3 y + z = 5 2 x - 3 y + z = - 2 - 4 x + 6 y - 2 z = 10 (I) (II) (III)

We can see that the left-hand sides of Equation (I) and Equation (II) are the same, while the right-hand side is different. This will be represented graphically as two parallel planes. Furthermore, notice that left-hand side of Equation (III) is a multiple of the left-hand side of the first two equations. Let's divide Equation (III) by

- 2 .

- 4 x + 6 y - 2 z = 10

DivEqn

$LHS / (- 2) = RHS / (- 2)$

2 x - 3 y + z = - 5

Let's rewrite our original system as the equivalent one shown below.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x - 3 y + z = 5 2 x - 3 y + z = - 2 2 x - 3 y + z = - 5 (I) (II) (III)

Notice that the equations forming the system are all planes that are parallel to each other.

Since they do not intersect at all, the system will not have any solutions. Therefore, this is an inconsistent system.

Exercise

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