dependent system
Has infinitely many solutions. The graph consists of two coincident lines.
A dependent system is a system of equations in which the equations are not sufficient to determine unique values for each variable. In such a system, the equations are either multiples of each other or completely identical. This leads to an infinite number of solutions or no solution at all.
For example, suppose we have two equations:
2x + 3y = 7
4x + 6y = 14
These equations are dependent because the second equation is simply twice the first equation. Therefore, when we try to solve the system, we end up with the same equation twice. This means that there are infinitely many solutions to the system, since any value of x and y that satisfies one equation will also satisfy the other.
Another example of a dependent system is:
x + 2y = 5
2x + 4y = 10
These equations are not independent, as the second equation is a multiple of the first equation. This means that both equations are essentially saying the same thing, and there is no way to uniquely solve for both variables. In this case, there are infinitely many solutions (in this case, any value of x + 2y = 5 will work).
To summarize, a dependent system occurs when the equations in a system are not independent and do not provide enough information to solve the variables uniquely.
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