To derive the equation of the line using the given points \((1, 1)\) and \((0, 4)\), we first need to calculate the slope of the line.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((1, 1)\) and \((0, 4)\), we can assign: \((x_1, y_1) = (1, 1)\) and \((x_2, y_2) = (0, 4)\).
Now substitute these values into the slope formula:
\[ m = \frac{4 - 1}{0 - 1} = \frac{3}{-1} = -3 \]
Now we have the slope \(m = -3\).
Next, we use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \]
We can use one of our points. Let's use the point \((0, 4)\):
\[ y - 4 = -3(x - 0) \]
This simplifies to:
\[ y - 4 = -3x \] \[ y = -3x + 4 \]
Now, we convert this into slope-intercept form (which is \(y = mx + b\)). Thus, we can rearrange the equation:
\[ y = -3x + 4 \]
Looking at the options provided, none of them exactly match our derived equation \(y = -3x + 4\). However, we can check if option (d) translates similarly since it is written in a different form:
Option d: 3x - y = 4
Rearranging it into slope-intercept form:
\[ -y = -3x + 4 \] \[ y = 3x - 4 \]
So, option \(d\) is not equivalent to the derived line, hence the correct slope-intercept form derived from the points is:
\[ \boxed{y = -3x + 4} \]
However, based on how this aligns with your given options, you may want to double-check, as none of the available answers reflect the correct equation we derived.