Reduce the equation to `sqrt(3)x+y+1=0` to the form `y=mx+c` and hence , find the slope the inclination to the x-axis and the intercept on the y-axis.

Answer

To reduce the equation \(\sqrt{3}x + y + 1 = 0\) to the form \(y = mx + c\) and find the slope of inclination to the x-axis and the y-intercept, follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ \sqrt{3}x + y + 1 = 0 \]
To isolate \(y\), subtract \(\sqrt{3}x\) and 1 from both sides:
\[ y = -\sqrt{3}x - 1 \]

### Step 2: Identifying the Slope and Intercept
Now, the equation is in the form \(y = mx + c\), where:
- \(m\) (the slope) is \(-\sqrt{3}\)
- \(c\) (the y-intercept) is \(-1\)

### Step 3: Finding the Slope of Inclination
The slope \(m\) can be related to the angle of inclination \(\theta\) with the x-axis using the formula:
\[ m = \tan(\theta) \]
Thus,
\[ \tan(\theta) = -\sqrt{3} \]
To find \(\theta\), we take the arctangent:
\[ \theta = \tan^{-1}(-\sqrt{3}) \]
The angle whose tangent is \(-\sqrt{3}\) corresponds to \(240^\circ\) or \(60^\circ\) in the third quadrant.

### Step 4: Conclusion
- The slope of the line is \(-\sqrt{3}\).
- The angle of inclination to the x-axis is \(240^\circ\) (or \(60^\circ\) in the negative direction).
- The y-intercept is \(-1\).

### Summary of Results
- Slope \(m = -\sqrt{3}\)
- Angle of inclination \(\theta = 240^\circ\)
- Y-intercept \(c = -1\)