Find the equation to the straight line
Cutting off an intercept unity from the positive direction of the axis of y and inclined at `45^(@)` to the axis of x.

Answer

To find the equation of the straight line that cuts off an intercept of unity from the positive direction of the y-axis and is inclined at \(45^\circ\) to the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
The line cuts the y-axis at \(y = 1\) (since the intercept is unity) and makes an angle of \(45^\circ\) with the x-axis.

2. **Finding the Slope:**
The slope \(m\) of a line inclined at an angle \(\theta\) to the x-axis can be calculated using the tangent function:
\[ m = \tan(\theta) \]
Given that \(\theta = 45^\circ\):
\[ m = \tan(45^\circ) = 1 \]

3. **Using the Slope-Intercept Form:**
The equation of a straight line in slope-intercept form is:
\[ y = mx + c \]
Substituting the value of \(m\):
\[ y = 1x + c \quad \text{or} \quad y = x + c \]

4. **Finding the y-Intercept:**
Since the line cuts the y-axis at \(y = 1\), we can substitute \(x = 0\) into the equation to find \(c\):
\[ y = x + c \quad \Rightarrow \quad 1 = 0 + c \quad \Rightarrow \quad c = 1 \]

5. **Final Equation of the Line:**
Now substituting \(c\) back into the equation:
\[ y = x + 1 \]

### Conclusion:
The equation of the straight line is:
\[ \boxed{y = x + 1} \]