The inclination of the line `x-y+3=0` with the positive direction of x-axis is

Options

A. `45^(@)`
B. `135^(@)`
C. `-45^(@)`
D. `-135^(@)`

Answer

To find the inclination of the line given by the equation \( x - y + 3 = 0 \) with the positive direction of the x-axis, we can follow these steps:

### Step 1: Rewrite the equation in slope-intercept form
We start with the equation of the line:
\[ x - y + 3 = 0 \]
Rearranging this equation to solve for \( y \):
\[ y = x + 3 \]

### Step 2: Identify the slope of the line
From the equation \( y = mx + c \), we can identify the slope \( m \). Here, the slope \( m \) is:
\[ m = 1 \]

### Step 3: Relate the slope to the angle of inclination
The slope of a line is related to the angle of inclination \( \theta \) with the positive x-axis by the formula:
\[ m = \tan(\theta) \]
Substituting the value of \( m \):
\[ 1 = \tan(\theta) \]

### Step 4: Calculate the angle of inclination
To find \( \theta \), we take the arctangent (inverse tangent) of both sides:
\[ \theta = \tan^{-1}(1) \]
The angle whose tangent is 1 is:
\[ \theta = 45^\circ \]

### Conclusion
Thus, the inclination of the line \( x - y + 3 = 0 \) with the positive direction of the x-axis is:
\[ \boxed{45^\circ} \]

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