What is the equation of the line whose y - intercept is `(-3)/(4)` and making an angle of ` 45^(@)` with the positive x - axis ?

Options

A. `4 x - 4y = 3 `
B. ` 4 x - 4 y = - 3 `
C. ` 3 x - 3y = 4`
D. ` 3 x - 3y = - 4 `

Answer

To find the equation of the line whose y-intercept is \(-\frac{3}{4}\) and which makes an angle of \(45^\circ\) with the positive x-axis, we can follow these steps:

### Step 1: Identify the y-intercept
The y-intercept \(c\) is given as \(-\frac{3}{4}\).

### Step 2: Determine the slope
The slope \(m\) of the line can be calculated using the angle \(\alpha\) that the line makes with the positive x-axis. The formula for the slope in terms of the angle is:
\[ m = \tan(\alpha) \]
Given that \(\alpha = 45^\circ\):
\[ m = \tan(45^\circ) = 1 \]

### Step 3: Write the equation of the line
The general form of the equation of a line is:
\[ y = mx + c \]
Substituting the values of \(m\) and \(c\):
\[ y = 1 \cdot x - \frac{3}{4} \]
This simplifies to:
\[ y = x - \frac{3}{4} \]

### Step 4: Rearranging to standard form
To express the equation in standard form, we can eliminate the fraction:
\[ y = x - \frac{3}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 4y = 4x - 3 \]
Rearranging gives:
\[ 4x - 4y = 3 \]

### Final Answer
The equation of the line is:
\[ 4x - 4y = 3 \]

---