Determine whether the system is inconsistent, dependent, or neither.
-4x + y + 2z = 1 -x + 4y + 2z= 5
-9x + 2y - 2z = 3
A) dependent
B) inconsistent
C) neither

Question

Quizplus · Accepted Answer

Adding the first two equations gives 3x + 5y + 4z = 6. If we multiply the second equation by 3 and add it to the first equation, we get the same equation. Therefore, the system is dependent. First, we can add the first and second equation to get 3x+5y+4z=6. This is a new equation that contains the same variables as the original system. Next, we can multiply the third equation by -1 and add it to the new equation to get:
3x+5y+4z=6
+9x-2y+2z=-3
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12x+3y+6z=3
We can simplify this equation by dividing everything by 3, giving 4x+y+2z=1. This equation is identical to the first equation in the original system! Therefore, we only have two distinct equations for three variables, which means that there are infinitely many solutions to this system. In other words, the system is dependent.

Determine Whether the System Is Inconsistent, Dependent, or Neither