What makes a system consistent? Inconsistent?
Consistent if there is at least 1 solution (there can be infinite) Inconsistent if there is no solution
A system is considered consistent if it has at least one solution, meaning that the system of equations can be solved for one or more values that allow all equations to be true simultaneously.
On the other hand, a system is considered inconsistent if it has no solutions, meaning that it is impossible for all equations in the system to be true simultaneously.
For example, consider the following system of linear equations:
2x + 3y = 7
4x + 6y = 14
We can determine if this system is consistent or inconsistent by solving it using elimination, substitution, or another method. In this case, if we divide the first equation by 2, we get:
x + (3/2)y = 7/2
Now, if we subtract this equation from the second equation, we get:
(3/2)y = 0
This implies that y must be 0. Substituting this value of y into the first equation, we get:
2x = 7
Thus, x is equal to 7/2. Therefore, this system has a unique solution of (7/2, 0), which means that it is consistent.
However, consider the following system of linear equations:
2x + 3y = 7
4x + 6y = 8
Using the same methods as before, we can find that there is no solution that satisfies both equations simultaneously. Therefore, this system is inconsistent.
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