For what value of k, the pair of equation `2x+ky=1,3x-5y=7` has a unique solution?

Options

A. `k=(-10)/(3)`
B. `kne(-10)/(3)`
C. `kne(10)/(3)`
D. `k ne(3)/(10)`.

Answer

To determine the value of \( k \) for which the pair of equations \( 2x + ky = 1 \) and \( 3x - 5y = 7 \) has a unique solution, we can follow these steps:

### Step 1: Write the equations in standard form
We can rewrite the equations in the standard form \( Ax + By + C = 0 \).

1. The first equation \( 2x + ky = 1 \) can be rewritten as:
\[ 2x + ky - 1 = 0 \]
Here, \( A_1 = 2 \), \( B_1 = k \), and \( C_1 = -1 \).

2. The second equation \( 3x - 5y = 7 \) can be rewritten as:
\[ 3x - 5y - 7 = 0 \]
Here, \( A_2 = 3 \), \( B_2 = -5 \), and \( C_2 = -7 \).

### Step 2: Use the condition for unique solutions
For a pair of linear equations to have a unique solution, the following condition must be satisfied:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]

Substituting the values we have:
\[ \frac{2}{3} \neq \frac{k}{-5} \]

### Step 3: Solve the inequality
To find the value of \( k \) that does not satisfy this condition, we can set up the equation:
\[ \frac{2}{3} = \frac{k}{-5} \]

Cross-multiplying gives:
\[ 2 \cdot (-5) = 3k \]
\[ -10 = 3k \]
\[ k = -\frac{10}{3} \]

### Step 4: Determine the condition for unique solution
Thus, for the equations to have a unique solution, \( k \) must not equal \( -\frac{10}{3} \).

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

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