Find the equation of the line through the points A(-1, 3) and B(0, 2). Hence, show that the points A, B and C(1, 1) are collinear. 

Answer

To find the equation of the line through the points A(-1, 3) and B(0, 2), and to show that the points A, B, and C(1, 1) are collinear, we can follow these steps:

### Step 1: Find the slope of the line (m)
The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

Here, we can take:
- A(-1, 3) as (x1, y1) = (-1, 3)
- B(0, 2) as (x2, y2) = (0, 2)

Substituting the values:
\[ m = \frac{2 - 3}{0 - (-1)} = \frac{-1}{1} = -1 \]

### Step 2: Use the point-slope form to find the equation of the line
The point-slope form of the equation of a line is:
\[ y - y1 = m(x - x1) \]

Using point A(-1, 3) and the slope m = -1:
\[ y - 3 = -1(x - (-1)) \]
\[ y - 3 = -1(x + 1) \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ y - 3 = -x - 1 \]
Adding 3 to both sides:
\[ y = -x - 1 + 3 \]
\[ y = -x + 2 \]

Rearranging gives us:
\[ x + y = 2 \]

### Step 4: Show that point C(1, 1) is on the line
To show that points A, B, and C are collinear, we need to check if point C(1, 1) satisfies the equation of the line we found.

Substituting C(1, 1) into the equation \( x + y = 2 \):
\[ 1 + 1 = 2 \]
Since the left-hand side (LHS) equals the right-hand side (RHS), point C satisfies the equation.

### Conclusion
Since point C lies on the line formed by points A and B, we conclude that points A, B, and C are collinear.

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