The system of equation
2x + y - 3z = 5
3x - 2y + 2z = 5
5x - 3y - z = 16

Options

A. (a)is inconsistent
B. (b)is consistent, with a unique solution
C. (c)is consistent, with infinitely many solutions
D. (d)has its solution lying aling x-axis in three dimensional space

Answer

To solve the given system of equations using determinants, we will follow these steps:

### Step 1: Write the system of equations in standard form
The given equations are:
1. \( 2x + y - 3z = 5 \)
2. \( 3x - 2y + 2z = 5 \)
3. \( 5x - 3y - z = 16 \)

### Step 2: Identify the coefficient matrix \( A \) and the constant matrix \( B \)
The coefficient matrix \( A \) is formed from the coefficients of \( x, y, z \) in the equations:
\[ A = \begin{bmatrix} 2 & 1 & -3 \\ 3 & -2 & 2 \\ 5 & -3 & -1 \end{bmatrix} \]

The constant matrix \( B \) is formed from the constants on the right-hand side of the equations:
\[ B = \begin{bmatrix} 5 \\ 5 \\ 16 \end{bmatrix} \]

### Step 3: Calculate the determinant of matrix \( A \)
To find the determinant of \( A \), we can use the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Where:
- \( a = 2, b = 1, c = -3 \)
- \( d = 3, e = -2, f = 2 \)
- \( g = 5, h = -3, i = -1 \)

Calculating the determinant:
\[ \text{det}(A) = 2((-2)(-1) - (2)(-3)) - 1((3)(-1) - (2)(5)) + (-3)((3)(-3) - (-2)(5)) \]
Calculating each term:
1. \( (-2)(-1) - (2)(-3) = 2 + 6 = 8 \)
2. \( (3)(-1) - (2)(5) = -3 - 10 = -13 \)
3. \( (3)(-3) - (-2)(5) = -9 + 10 = 1 \)

Substituting back:
\[ \text{det}(A) = 2(8) - 1(-13) - 3(1) \]
\[ = 16 + 13 - 3 = 26 \]

### Step 4: Analyze the determinant
Since \( \text{det}(A) = 26 \) which is not equal to 0, this implies that the system of equations is consistent and has a unique solution.

### Conclusion
The system of equations is consistent with a unique solution.

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