9514 1404 393
Step-by-step explanation:
Using a random number generator to choose coefficients, we can substitute the required solution to find the constants in the equations. Here is one possible set of equations.
17x +7y +8z = 6
12x -6y +z = -16
14x +8y +11z = 16
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Additional comment
"Standard form" requires the x-coefficient be positive. Truth be told, the random number generator came up with negative numbers for the first and last equations (on the second try). On the first try, it came up with a couple of zero coefficients. Such is the behavior of random numbers.
The trick with choosing a set of numbers "by hand" is doing it in such a way that the equations are independent.