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A system of equations with infinitely many solutions occurs when the equations in the system are equivalent.
This means that one equation can be derived from the other by algebraic manipulation. In this case, any values that satisfy one of the equations will also satisfy the other equation, so there are an infinite number of solutions.
To determine if a system of equations has infinitely many solutions, you can try to solve the system using one of the methods for solving systems of equations, such as graphing, substitution, or elimination. If you find that one of the equations can be derived from the other by algebraic manipulation, then the system has infinitely many solutions.
As for the system of equations 2x + y = 1 and 4x + 2y = 2, it has infinitely many solutions. This can be seen by solving the system using substitution. Solving the first equation for y gives y = 1 - 2x. Substituting this expression for y into the second equation gives 4x + 2(1 - 2x) = 2, which simplifies to 2x = 0. Thus, x = 0, and substituting this value back into the first equation gives y = 1 - 2(0) = 1. So one solution to the system is (x, y) = (0, 1). However, since the system has infinitely many solutions, there are an infinite number of other solutions as well, obtained by taking any values of x and y that satisfy the first equation.
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