Prove that the points A(1, 4), B(3, -2) and C(4, -5) are collinear. Also find the equation of the line on which these points lie.

Answer

To prove that the points A(1, 4), B(3, -2), and C(4, -5) are collinear, we will calculate the slopes of the line segments AB and BC. If the slopes are equal, then the points are collinear. We will also find the equation of the line on which these points lie.

### Step 1: Calculate the slope of line segment AB
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:

\[ m = \frac{y2 - y1}{x2 - x1} \]

For points A(1, 4) and B(3, -2):

- \(x_1 = 1\), \(y_1 = 4\)
- \(x_2 = 3\), \(y_2 = -2\)

Substituting these values into the slope formula:

\[ m_{AB} = \frac{-2 - 4}{3 - 1} = \frac{-6}{2} = -3 \]

### Step 2: Calculate the slope of line segment BC
Now, we calculate the slope between points B(3, -2) and C(4, -5):

For points B(3, -2) and C(4, -5):

- \(x_1 = 3\), \(y_1 = -2\)
- \(x_2 = 4\), \(y_2 = -5\)

Substituting these values into the slope formula:

\[ m_{BC} = \frac{-5 - (-2)}{4 - 3} = \frac{-5 + 2}{1} = \frac{-3}{1} = -3 \]

### Step 3: Compare the slopes
Since \(m_{AB} = -3\) and \(m_{BC} = -3\), we have:

\[ m_{AB} = m_{BC} \]

This implies that the points A, B, and C are collinear.

### Step 4: Find the equation of the line
To find the equation of the line, we can use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

We can use point A(1, 4) and the slope \(m = -3\):

Substituting into the equation:

\[ y - 4 = -3(x - 1) \]

Expanding this:

\[ y - 4 = -3x + 3 \]

Rearranging gives:

\[ 3x + y - 7 = 0 \]

Thus, the equation of the line is:

\[ 3x - y - 7 = 0 \]

### Final Result
The points A(1, 4), B(3, -2), and C(4, -5) are collinear, and the equation of the line on which these points lie is:

\[ 3x - y - 7 = 0 \]

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