The equation `2x+y=5, x+3y =5, x-2y=0` have :

Options

A. no solution
B. one solution
C. two solutions
D. infinitely many solutions

Answer

To determine the number of solutions for the given equations \(2x + y = 5\), \(x + 3y = 5\), and \(x - 2y = 0\), we will use the method of ranks of matrices.

### Step 1: Write the equations in matrix form
We can express the system of equations in the form \(AX = B\), where:
\[ A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \\ 1 & -2 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 5 \\ 0 \end{bmatrix} \]

### Step 2: Find the rank of matrix \(A\)
To find the rank of matrix \(A\), we will perform row operations to reduce it to row echelon form.

1. Start with the original matrix:
\[ A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \\ 1 & -2 \end{bmatrix} \]

2. We can perform the following row operations:
- \(R_2 \leftarrow R_2 - \frac{1}{2}R_1\)
- \(R_3 \leftarrow R_3 - \frac{1}{2}R_1\)

This gives us:
\[ \begin{bmatrix} 2 & 1 \\ 0 & \frac{5}{2} \\ 0 & -\frac{5}{2} \end{bmatrix} \]

3. Now, we can further simplify \(R_3\):
- \(R_3 \leftarrow R_3 + R_2\)

This results in:
\[ \begin{bmatrix} 2 & 1 \\ 0 & \frac{5}{2} \\ 0 & 0 \end{bmatrix} \]

From this, we can see that there are 2 non-zero rows, so the rank of matrix \(A\) is \(2\).

### Step 3: Find the rank of the augmented matrix \([A|B]\)
Now we form the augmented matrix \([A|B]\):
\[ [A|B] = \begin{bmatrix} 2 & 1 & | & 5 \\ 1 & 3 & | & 5 \\ 1 & -2 & | & 0 \end{bmatrix} \]

1. We perform similar row operations on the augmented matrix:
- \(R_2 \leftarrow R_2 - \frac{1}{2}R_1\)
- \(R_3 \leftarrow R_3 - \frac{1}{2}R_1\)

This gives us:
\[ \begin{bmatrix} 2 & 1 & | & 5 \\ 0 & \frac{5}{2} & | & \frac{15}{2} \\ 0 & -\frac{5}{2} & | & -\frac{5}{2} \end{bmatrix} \]

2. Now, we can simplify \(R_3\):
- \(R_3 \leftarrow R_3 + R_2\)

This results in:
\[ \begin{bmatrix} 2 & 1 & | & 5 \\ 0 & \frac{5}{2} & | & \frac{15}{2} \\ 0 & 0 & | & 0 \end{bmatrix} \]

Again, we see that there are 2 non-zero rows, so the rank of the augmented matrix \([A|B]\) is also \(2\).

### Step 4: Compare ranks
Now we compare the ranks:
- Rank of \(A\) = 2
- Rank of \([A|B]\) = 2
- Number of unknowns = 2

Since the rank of \(A\) is equal to the rank of \([A|B]\) and both are equal to the number of unknowns, we conclude that the system has a unique solution.

### Final Answer
The equations \(2x + y = 5\), \(x + 3y = 5\), and \(x - 2y = 0\) have **one unique solution**.

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