In relation to the line : 7(x-2) =5(y+3), the point (3,-2) lies on

Options

A. the line
B. origin side of the line
C. non-origin side of the line
D. none of these

Answer

To determine the position of the point (3, -2) in relation to the line given by the equation \(7(x-2) = 5(y+3)\), we will follow these steps:

### Step 1: Rewrite the line equation in standard form
The given equation is:
\[ 7(x - 2) = 5(y + 3) \]
Expanding both sides:
\[ 7x - 14 = 5y + 15 \]
Rearranging gives:
\[ 7x - 5y - 29 = 0 \]
This can be expressed in the form \(Ax + By + C = 0\) where \(A = 7\), \(B = -5\), and \(C = -29\).

### Step 2: Substitute the point (3, -2) into the line equation
We will substitute \(x = 3\) and \(y = -2\) into the equation \(7x - 5y - 29 = 0\):
\[ 7(3) - 5(-2) - 29 = 0 \]
Calculating this:
\[ 21 + 10 - 29 = 0 \]
\[ 31 - 29 = 2 \]
Thus, we get:
\[ 2 \neq 0 \]

### Step 3: Determine the position of the point
Since the result is not equal to zero, the point (3, -2) does not lie on the line. Next, we will determine which side of the line the point lies on.

### Step 4: Evaluate the expression for the point (3, -2)
We will evaluate the expression \(7x - 5y - 29\) using the coordinates of the point (3, -2):
\[ 7(3) - 5(-2) - 29 = 2 \]
Since \(2 > 0\), the point (3, -2) lies on the side of the line where the expression is positive.

### Conclusion
The point (3, -2) lies on the side of the line defined by the equation \(7(x-2) = 5(y+3)\) where the expression evaluates to a positive value.