The equations `x+2y+3z=1, 2x+y+3z=2,5x+5y+9z=4` have

Options

A. unique solutions
B. infinite many solutions
C. inconsistent
D. None of these

Answer

To determine the number of solutions for the given system of equations:

1. **Write the equations in matrix form**:
The given equations are:
\[ \begin{align*} x + 2y + 3z &= 1 \quad (1) \\ 2x + y + 3z &= 2 \quad (2) \\ 5x + 5y + 9z &= 4 \quad (3) \end{align*} \]

We can represent this system in the form \(Ax = b\), where:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad b = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} \]

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 is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \(A\):
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9 \end{pmatrix} \]
We can calculate the determinant as follows:
\[ \text{det}(A) = 1 \cdot (1 \cdot 9 - 3 \cdot 5) - 2 \cdot (2 \cdot 9 - 3 \cdot 5) + 3 \cdot (2 \cdot 5 - 1 \cdot 5) \]
Simplifying each term:
\[ = 1 \cdot (9 - 15) - 2 \cdot (18 - 15) + 3 \cdot (10 - 5) \]
\[ = 1 \cdot (-6) - 2 \cdot 3 + 3 \cdot 5 \]
\[ = -6 - 6 + 15 \]
\[ = 3 \]

3. **Determine the nature of the solutions**:
Since \(\text{det}(A) \neq 0\), the system of equations has a unique solution.

**Final Conclusion**:
The equations \(x + 2y + 3z = 1\), \(2x + y + 3z = 2\), and \(5x + 5y + 9z = 4\) have a unique solution.

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