Answer:
Step-by-step explanation:
To know the type of the given system, we need to solve it.
The given system of equations is
[tex]\left \{ {{2x-31y=-2} \atop {2x+2y=42}} \right.[/tex]
We can solve this system by multiplying the first equation with -1, and subtracting after
[tex]\left \{ {{-2x+31y=+2} \atop {2x+2y=42}} \right.\\\\33y=44\\y=\frac{44}{33}\\ y=\frac{4}{3}[/tex]
Then, we use this value to find the other one
[tex]2x+2y=42\\2x+2\frac{4}{3}=42\\ 2x=42-\frac{8}{3}\\ 2x=\frac{126-8}{3} \\2x=\frac{118}{3} \\x=\frac{118}{6}\\ x=\frac{59}{3}[/tex]
As you observe, the system has a solution, which is [tex](\frac{4}{3}, \frac{59}{3})[/tex]. That means the system cannot be inconsistent. because it has a solution. Remember that inconsitent systems refer to those that don't have any solution, that is, the lines are parallel.
On the other hand, independent systems are those where each equation produces a different line, and the solution is the interception of those lines, like this case. We have a solution and two different lines represented by each linear equation.
Therefore, the right answer is independent.
