How many solutions does the following system of linear equations have ? -x +…

System of Equations:

1. -x + 5y = -1

2. x - y = 2

3. x + 3y = 3

Step 1: Write the Augmented Matrix [A|B]

[ -1 5 | -1 ]

[ 1 -1 | 2 ]

[ 1 3 | 3 ]

Step 2: Forward Elimination to Row Echelon Form (Gaussian Elimination)

First, swap Row 1 (R1) and Row 2 (R2) to get a 1 as the top-left pivot:

R1 <--> R2

[ 1 -1 | 2 ]

[ -1 5 | -1 ]

[ 1 3 | 3 ]

Now, eliminate the entries below the first pivot in Column 1:

• To eliminate the -1 in Row 2: R2 = R2 + R1

• To eliminate the 1 in Row 3: R3 = R3 - R1

[ 1 -1 | 2 ]

[ 0 4 | 1 ]

[ 0 4 | 1 ]

Next, eliminate the entry below the pivot in Column 2 by subtracting Row 2 from Row 3:

R3 = R3 - R2

[ 1 -1 | 2 ]

[ 0 4 | 1 ]

[ 0 0 | 0 ]

Step 3: Analyze the Row Echelon Form

Looking at the reduced matrix:

• Number of variables (x, y) = 2

• Rank of the coefficient matrix A (number of non-zero rows) = 2

• Rank of the augmented matrix [A|B] = 2

Since the bottom row turned completely into zeros (0 = 0), the system is consistent and does not contain any contradictions.

Because Rank(A) = Rank([A|B]) = Number of variables, the system has exactly one unique solution.

Step 4: Back-Substitution (Optional verification)

From Row 2: 4y = 1 --> y = 1/4

From Row 1: x - y = 2 --> x - 1/4 = 2 --> x = 9/4

Final Answer:

The system of linear equations has exactly one unique solution.