Find the equation of a line wiith slope -1 and cutting off an intercept of 4 units on negative direction of
(a) x-axis
(b) y-axis

Answer

To find the equation of a line with a slope of -1 and intercepts of 4 units on the negative direction of the x-axis and y-axis, we can follow these steps:

### Part (a): Finding the equation when the intercept is on the x-axis

1. **Identify the x-intercept**: Since the line cuts off an intercept of 4 units on the negative x-axis, the x-intercept is at the point (-4, 0).

2. **Use the slope**: The slope (m) of the line is given as -1.

3. **Use the point-slope form of the equation of a line**: The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line.

4. **Substitute the values**: Here, \((x_1, y_1) = (-4, 0)\) and \(m = -1\):
\[ y - 0 = -1(x - (-4)) \]
This simplifies to:
\[ y = -1(x + 4) \]

5. **Distribute and rearrange**:
\[ y = -x - 4 \]
Rearranging gives:
\[ x + y + 4 = 0 \]

### Part (b): Finding the equation when the intercept is on the y-axis

1. **Identify the y-intercept**: Since the line cuts off an intercept of 4 units on the negative y-axis, the y-intercept is at the point (0, -4).

2. **Use the slope**: The slope (m) of the line remains -1.

3. **Use the point-slope form of the equation of a line**: Again, we use:
\[ y - y_1 = m(x - x_1) \]

4. **Substitute the values**: Here, \((x_1, y_1) = (0, -4)\) and \(m = -1\):
\[ y - (-4) = -1(x - 0) \]
This simplifies to:
\[ y + 4 = -x \]

5. **Rearranging gives**:
\[ x + y + 4 = 0 \]

### Final Result
For both parts (a) and (b), the equation of the line is:
\[ x + y + 4 = 0 \]