Find the coordinates of the point of intersection of x - 3y = 0 and the line segment joining the points (- 2, - 5) and (6, 3).

Answer

To find the coordinates of the point of intersection of the line \( x - 3y = 0 \) and the line segment joining the points \((-2, -5)\) and \((6, 3)\), we can follow these steps:

### Step 1: Write the equation of the line segment joining the points \((-2, -5)\) and \((6, 3)\).

The formula for the equation of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]

Here, let \((x_1, y_1) = (-2, -5)\) and \((x_2, y_2) = (6, 3)\).

Substituting the values, we get:

\[ \frac{y - (-5)}{3 - (-5)} = \frac{x - (-2)}{6 - (-2)} \]

This simplifies to:

\[ \frac{y + 5}{8} = \frac{x + 2}{8} \]

Cross-multiplying gives:

\[ y + 5 = x + 2 \]

Rearranging this, we find the equation of the line segment:

\[ y = x - 3 \]

### Step 2: Solve the equations simultaneously.

Now we have two equations:
1. \( x - 3y = 0 \) (Equation of the first line)
2. \( y = x - 3 \) (Equation of the line segment)

Substituting the second equation into the first:

\[ x - 3(x - 3) = 0 \]

Expanding this gives:

\[ x - 3x + 9 = 0 \]

Combining like terms results in:

\[ -2x + 9 = 0 \]

### Step 3: Solve for \(x\).

Rearranging gives:

\[ 2x = 9 \implies x = \frac{9}{2} \]

### Step 4: Substitute \(x\) back to find \(y\).

Now substitute \(x = \frac{9}{2}\) back into the equation \(y = x - 3\):

\[ y = \frac{9}{2} - 3 = \frac{9}{2} - \frac{6}{2} = \frac{3}{2} \]

### Final Coordinates

Thus, the coordinates of the point of intersection are:

\[ \left( \frac{9}{2}, \frac{3}{2} \right) \]