Find the value(s) of p for the following pair of equations:
`-3x +5y=7 and 2px -3y=1`, if the lines represented by these equations are intersecting at a unique point.

Answer

To find the value(s) of \( p \) for which the pair of equations \( -3x + 5y = 7 \) and \( 2px - 3y = 1 \) intersect at a unique point, we can follow these steps:

### Step 1: Write the equations in standard form
The standard form of a linear equation is \( Ax + By + C = 0 \).

For the first equation:
\[ -3x + 5y - 7 = 0 \]
Here, \( A_1 = -3 \), \( B_1 = 5 \), and \( C_1 = -7 \).

For the second equation:
\[ 2px - 3y - 1 = 0 \]
Here, \( A_2 = 2p \), \( B_2 = -3 \), and \( C_2 = -1 \).

### Step 2: Use the condition for unique intersection
For the lines to intersect at a unique point, the condition is:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]

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

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ -3 \cdot (-3) \neq 5 \cdot (2p) \]
This simplifies to:
\[ 9 \neq 10p \]

### Step 4: Solve for \( p \)
From the inequality \( 9 \neq 10p \), we can express \( p \):
\[ p \neq \frac{9}{10} \]

### Conclusion
The value of \( p \) must not equal \( \frac{9}{10} \) for the lines to intersect at a unique point.