Does the line `3x - 5y = 6` bisect the join of (5,-2) and (-1, 2) ? 

Answer

To determine whether the line \(3x - 5y = 6\) bisects the line segment joining the points \(A(5, -2)\) and \(B(-1, 2)\), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment AB
The midpoint \(M\) of the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \(A(5, -2)\) and \(B(-1, 2)\).

Substituting the coordinates:
\[ M = \left( \frac{5 + (-1)}{2}, \frac{-2 + 2}{2} \right) = \left( \frac{4}{2}, \frac{0}{2} \right) = (2, 0) \]

### Step 2: Check if the Midpoint M Lies on the Line
Next, we need to check if the midpoint \(M(2, 0)\) satisfies the equation of the line \(3x - 5y = 6\).

Substituting \(x = 2\) and \(y = 0\) into the equation:
\[ 3(2) - 5(0) = 6 \]
Calculating this gives:
\[ 6 - 0 = 6 \]
Since this is true, we conclude that:
\[ 6 = 6 \]
Thus, the point \(M(2, 0)\) lies on the line \(3x - 5y = 6\).

### Conclusion
Since the midpoint \(M(2, 0)\) lies on the line \(3x - 5y = 6\), we can conclude that the line bisects the segment joining points \(A(5, -2)\) and \(B(-1, 2)\).

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