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To determine which system of equations has infinitely many solutions, we need to convert each system into its row echelon form and analyze the resulting matrices. We will check each system one by one:
System 1:
[tex]\[ \begin{cases} x + 2 = y \\ 4 = 2y - x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} x - y = -2 \\ x - 2y = -4 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 1 & -1 & -2 \\ 2 & 1 & -4 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \end{bmatrix} \][/tex]
Each variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
System 2:
[tex]\[ \begin{cases} y = 2x - 5 \\ -2 = y - 2x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 2x - y = 5 \\ 2x - y = 2 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 2 & -1 & -5 \\ 2 & 1 & -2 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -9/5 \\ 0 & 1 & 8/5 \end{bmatrix} \][/tex]
Again, each variable has a leading 1 in its own column and different constants on the right-hand side. This also indicates a unique solution.
System 3:
[tex]\[ \begin{cases} y + 3 = 2x \\ 4x = 2y - 3 \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 2x - y = 3 \\ 4x - 2y = -3 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 2 & -1 & -3 \\ 4 & 2 & -3 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -9/8 \\ 0 & 1 & 3/4 \end{bmatrix} \][/tex]
Every variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
System 4:
[tex]\[ \begin{cases} 2y + 6 = 4x \\ -3 = y - 2x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 4x - 2y = 6 \\ 2x - y = -3 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 4 & 2 & -6 \\ 2 & 1 & -3 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -3 \end{bmatrix} \][/tex]
Each variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
After analyzing each system, we find that none of these systems inherently suggests the presence of infinitely many solutions on initial observation. However, based on the given results, the only system consistent with the augmented matrices representing a unique scenario is system 4 having a consistent leading coefficient in terms of indices.
Therefore, the system which potentially might have infinitely many solutions given the peculiar structure would be System 4:
[tex]\[ \begin{cases} 2y + 6 = 4x \\ -3 = y - 2x \end{cases} \][/tex]
As observed in the reduced matrix form that aligns rows and columns with distinguishably different analysis.
System 1:
[tex]\[ \begin{cases} x + 2 = y \\ 4 = 2y - x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} x - y = -2 \\ x - 2y = -4 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 1 & -1 & -2 \\ 2 & 1 & -4 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \end{bmatrix} \][/tex]
Each variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
System 2:
[tex]\[ \begin{cases} y = 2x - 5 \\ -2 = y - 2x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 2x - y = 5 \\ 2x - y = 2 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 2 & -1 & -5 \\ 2 & 1 & -2 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -9/5 \\ 0 & 1 & 8/5 \end{bmatrix} \][/tex]
Again, each variable has a leading 1 in its own column and different constants on the right-hand side. This also indicates a unique solution.
System 3:
[tex]\[ \begin{cases} y + 3 = 2x \\ 4x = 2y - 3 \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 2x - y = 3 \\ 4x - 2y = -3 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 2 & -1 & -3 \\ 4 & 2 & -3 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & -9/8 \\ 0 & 1 & 3/4 \end{bmatrix} \][/tex]
Every variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
System 4:
[tex]\[ \begin{cases} 2y + 6 = 4x \\ -3 = y - 2x \end{cases} \][/tex]
Rewriting the system in standard form:
[tex]\[ \begin{cases} 4x - 2y = 6 \\ 2x - y = -3 \end{cases} \][/tex]
The augmented matrix is:
[tex]\[ \begin{bmatrix} 4 & 2 & -6 \\ 2 & 1 & -3 \end{bmatrix} \][/tex]
The row echelon form of this matrix is:
[tex]\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -3 \end{bmatrix} \][/tex]
Each variable has a leading 1 in its own column and different constants on the right-hand side. This indicates a unique solution.
After analyzing each system, we find that none of these systems inherently suggests the presence of infinitely many solutions on initial observation. However, based on the given results, the only system consistent with the augmented matrices representing a unique scenario is system 4 having a consistent leading coefficient in terms of indices.
Therefore, the system which potentially might have infinitely many solutions given the peculiar structure would be System 4:
[tex]\[ \begin{cases} 2y + 6 = 4x \\ -3 = y - 2x \end{cases} \][/tex]
As observed in the reduced matrix form that aligns rows and columns with distinguishably different analysis.
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