Find the equation of a line passing through the points `(7, -3) and (2, -2)`. If this line meets x-axis at point P and y-axis at point Q, find the co-ordinates of points P and Q.

Answer

To find the equation of the line passing through the points (7, -3) and (2, -2), and to determine where this line intersects the x-axis and y-axis, we can follow these steps:

### Step 1: Identify the points
Let the points be:
- Point A (7, -3) = (x1, y1)
- Point B (2, -2) = (x2, y2)

### Step 2: Calculate the slope (m) of the line
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is:
\[ m = \frac{y2 - y1}{x2 - x1} \]

Substituting the values:
\[ m = \frac{-2 - (-3)}{2 - 7} = \frac{-2 + 3}{2 - 7} = \frac{1}{-5} = -\frac{1}{5} \]

### Step 3: Use the point-slope form to find the equation of the line
The point-slope form of the equation of a line is given by:
\[ y - y1 = m(x - x1) \]

Using point A (7, -3) and the slope we calculated:
\[ y - (-3) = -\frac{1}{5}(x - 7) \]
\[ y + 3 = -\frac{1}{5}x + \frac{7}{5} \]

### Step 4: Rearranging to slope-intercept form
Now, we rearrange the equation:
\[ y = -\frac{1}{5}x + \frac{7}{5} - 3 \]
Convert -3 to a fraction:
\[ -3 = -\frac{15}{5} \]
So,
\[ y = -\frac{1}{5}x + \frac{7}{5} - \frac{15}{5} \]
\[ y = -\frac{1}{5}x - \frac{8}{5} \]

### Step 5: Find the x-intercept (Point P)
To find the x-intercept, set y = 0:
\[ 0 = -\frac{1}{5}x - \frac{8}{5} \]
Multiply through by -5 to eliminate the fraction:
\[ 0 = x + 8 \]
Thus,
\[ x = -8 \]
So, the coordinates of point P are (-8, 0).

### Step 6: Find the y-intercept (Point Q)
To find the y-intercept, set x = 0:
\[ y = -\frac{1}{5}(0) - \frac{8}{5} \]
\[ y = -\frac{8}{5} \]
So, the coordinates of point Q are (0, -8/5).

### Final Answer
The coordinates of point P are (-8, 0) and the coordinates of point Q are (0, -8/5).

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