If the system of linear equations `x-4y+7z=g`, `3y-5z=h`, `-2x+5y-9z=k` is consistent, then

Options

A. g + h + k = 0
B. 2g + h + k = 0
C. g + h + 2k = 0
D. g + 2h + k = 0

Answer

To determine the conditions under which the system of linear equations is consistent, we need to analyze the given equations:

1. \( x - 4y + 7z = g \) (Equation 1)
2. \( 3y - 5z = h \) (Equation 2)
3. \( -2x + 5y - 9z = k \) (Equation 3)

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

We can represent the system of equations in matrix form as:

\[ \begin{bmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} g \\ h \\ k \end{bmatrix} \]

### Step 2: Find the determinant of the coefficient matrix

To check for consistency, we need to find the determinant of the coefficient matrix:

\[ \Delta = \begin{vmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{vmatrix} \]

Calculating the determinant:

\[ \Delta = 1 \cdot \begin{vmatrix} 3 & -5 \\ 5 & -9 \end{vmatrix} - (-4) \cdot \begin{vmatrix} 0 & -5 \\ -2 & -9 \end{vmatrix} + 7 \cdot \begin{vmatrix} 0 & 3 \\ -2 & 5 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 3 & -5 \\ 5 & -9 \end{vmatrix} = (3)(-9) - (-5)(5) = -27 + 25 = -2 \)

2. \( \begin{vmatrix} 0 & -5 \\ -2 & -9 \end{vmatrix} = (0)(-9) - (-5)(-2) = 0 - 10 = -10 \)

3. \( \begin{vmatrix} 0 & 3 \\ -2 & 5 \end{vmatrix} = (0)(5) - (3)(-2) = 0 + 6 = 6 \)

Now substituting back into the determinant calculation:

\[ \Delta = 1(-2) + 4(-10) + 7(6) = -2 - 40 + 42 = 0 \]

### Step 3: Set up the augmented matrix and find the conditions for consistency

Next, we need to find the conditions for the augmented matrix:

\[ \Delta_1 = \begin{vmatrix} g & -4 & 7 \\ h & 3 & -5 \\ k & 5 & -9 \end{vmatrix} \]

We will calculate this determinant similarly:

\[ \Delta_1 = g \cdot \begin{vmatrix} 3 & -5 \\ 5 & -9 \end{vmatrix} - (-4) \cdot \begin{vmatrix} h & -5 \\ k & -9 \end{vmatrix} + 7 \cdot \begin{vmatrix} h & 3 \\ k & 5 \end{vmatrix} \]

### Step 4: Solve for the conditions

To ensure the system is consistent, we need:

\[ \Delta_1 = 0 \]

This leads to a set of equations based on the determinants calculated. Solving these will give us the conditions on \(g\), \(h\), and \(k\) for the system to be consistent.

### Final Conditions

After solving the equations, we find:

1. \( -2g - h - k = 0 \)
2. \( 2g + h + k = 0 \)

### Summary

Thus, the conditions for the system of equations to be consistent are:

\[ -2g - h - k = 0 \quad \text{and} \quad 2g + h + k = 0 \]