Does the line `3x = y + 1` bisect the line segment joining A (-2, 3) and B (4, 1)?

Answer

To determine whether the line \(3x = y + 1\) bisects the line segment joining points A (-2, 3) and B (4, 1), we will follow these steps:

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

For points \(A(-2, 3)\) and \(B(4, 1)\):
- \(x_1 = -2\), \(y_1 = 3\)
- \(x_2 = 4\), \(y_2 = 1\)

Calculating the midpoint:
\[ M = \left(\frac{-2 + 4}{2}, \frac{3 + 1}{2}\right) = \left(\frac{2}{2}, \frac{4}{2}\right) = (1, 2) \]

### Step 2: Rewrite the Line Equation
The given line equation is:
\[ 3x = y + 1 \]
We can rearrange it into standard form:
\[ 3x - y - 1 = 0 \]

### Step 3: Check if the Midpoint Lies on the Line
To check if the line bisects the segment, we need to see if the midpoint \(M(1, 2)\) satisfies the line equation. Substitute \(x = 1\) and \(y = 2\) into the equation:
\[ 3(1) - 2 - 1 = 0 \]

Calculating:
\[ 3 - 2 - 1 = 0 \]
This simplifies to:
\[ 0 = 0 \]
Since this is true, the point \(M(1, 2)\) lies on the line.

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

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