For the system of equaltions :
`x+2y+3z=1`
`2x+y+3z=2`
`5x+5y+9z=4`

Options

A. there is only one solution
B. there exists infinitely many solution
C. there is no solution
D. none of these

Answer

To solve the system of equations given by:

1. \( x + 2y + 3z = 1 \)
2. \( 2x + y + 3z = 2 \)
3. \( 5x + 5y + 9z = 4 \)

we will represent the system in matrix form and then calculate the determinant to determine the nature of the solutions.

### Step 1: Write the system in matrix form

We can represent the system of equations in the form of a matrix \( A \) and a vector \( B \):

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} \]

### Step 2: Calculate the determinant of matrix \( A \)

To find the determinant of matrix \( A \), we use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where the matrix elements are:

\[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]

For our matrix \( A \):

- \( a = 1, b = 2, c = 3 \)
- \( d = 2, e = 1, f = 3 \)
- \( g = 5, h = 5, i = 9 \)

Now, we can calculate the determinant:

\[ \text{det}(A) = 1(1 \cdot 9 - 3 \cdot 5) - 2(2 \cdot 9 - 3 \cdot 5) + 3(2 \cdot 5 - 1 \cdot 5) \]

Calculating each term:

1. \( 1(9 - 15) = 1(-6) = -6 \)
2. \( -2(18 - 15) = -2(3) = -6 \)
3. \( 3(10 - 5) = 3(5) = 15 \)

Adding these together:

\[ \text{det}(A) = -6 - 6 + 15 = 3 \]

### Step 3: Analyze the determinant

Since \( \text{det}(A) = 3 \) which is not equal to zero, we conclude that the system of equations has a unique solution.

### Final Conclusion

The correct option is that there is only one solution to the system of equations.

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