For what value of K, the pair of equations `2x - 3y = 1 and kx + 5y = 7` has a unique solution?

Options

A. `k != 10/3`
B. `k != (-10)/3`
C. `k != (10)/(-3)`
D. None of these

Answer

To determine the value of \( K \) for which the pair of equations \( 2x - 3y = 1 \) and \( kx + 5y = 7 \) has a unique solution, we can use the condition for a unique solution of a system of linear equations.

### Step 1: Identify the coefficients
The given equations can be rewritten in the standard form \( Ax + By = C \):
1. \( 2x - 3y = 1 \) can be written as \( 2x + (-3)y = 1 \) where \( A_1 = 2, B_1 = -3, C_1 = 1 \).
2. \( kx + 5y = 7 \) can be written as \( kx + 5y = 7 \) where \( A_2 = k, B_2 = 5, C_2 = 7 \).

### Step 2: Apply the condition for a unique solution
For the system of equations to have a unique solution, the ratio of the coefficients of \( x \) and \( y \) must not be equal:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]
Substituting the coefficients:
\[ \frac{2}{k} \neq \frac{-3}{5} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 2 \cdot 5 \neq -3 \cdot k \]
This simplifies to:
\[ 10 \neq -3k \]

### Step 4: Solve for \( k \)
Rearranging the inequality, we get:
\[ -3k \neq 10 \]
Dividing both sides by -3 (and flipping the inequality sign):
\[ k \neq -\frac{10}{3} \]

### Final Answer
Thus, the value of \( K \) for which the pair of equations has a unique solution is:
\[ k \neq -\frac{10}{3} \]