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The matrix of coefficients for this system of linear equations has a rank of 2. The augmented matrix (including the constants on the right-hand side) also has a rank of 2. Since the rank of both the coefficient matrix and the augmented matrix is the same, but the rank is less than the number of variables (4), this system has **infinitely many solutions**.

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Here are 5 related questions to expand on this:

1. How do we interpret the rank of a matrix in the context of linear systems?
2. What is the difference between consistent and inconsistent systems of equations?
3. How can we find the general solution for a system with infinitely many solutions?
4. What is the geometric interpretation of systems with infinitely many solutions in 3D or 4D space?
5. How do matrix operations, such as Gaussian elimination, help solve systems of equations?

**Tip**: When solving systems of linear equations, always compare the rank of the coefficient matrix to the augmented matrix to determine if a system has no solutions, one solution, or infinitely many.

Determine whether the system of linear equations has 0, 1, or infinitely many solutions. x₁ + 2x₂ + x₄ = 7; x₁ + x₂ + x₃ - x₄ = 3; 3x₁ + x₂ + 5x₃ - 7x₄ = 1

Explore how to determine if a system of linear equations has 0, 1, or infinitely many solutions using matrix rank and Gaussian elimination. This guide walks through the step-by-step process of solving the system: x₁ + 2x₂ + x₄ = 7, x₁ + x₂ + x₃ - x₄ = 3, 3x₁ + x₂ + 5x₃ - 7x₄ = 1.

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