Answer:
[tex]\displaystyle y = \frac{1}{4}\, x + 2[/tex].
Step-by-step explanation:
The question is asking for the slope-intercept equation of a line. The slope-intercept form equation of a line is in the form [tex]y = m\, x + b[/tex], where:
For example, the equation of the first line, [tex]y = -4\, x + 1[/tex], is given in slope-intercept form. The slope of that line is [tex]m_{1} = (-4)[/tex] while the [tex]y[/tex]-intercept of that line is [tex]b_{1} = 1[/tex].
Both [tex]m[/tex] and [tex]b[/tex] need to be found for the second line.
Start by finding the slope [tex]m_{2}[/tex] of the second line.
Make use of the fact that if two lines in a plane are perpendicular to one another, the product of their slopes would be [tex](-1)[/tex]. In other words, if a line with slope [tex]m_{1}[/tex] is perpendicular to a line with slope [tex]m_{2}[/tex], then [tex]m_{1}\, m_{2} = (-1)[/tex]. Rearrange to obtain [tex]m_{2} = (-1) / m_{1}[/tex].
In this question, since the slope of the first line is [tex]m_{1} = (-4)[/tex], the slope of the second line (perpendicular to the first line) would be [tex]m_{2} = (-1) / (-4) = (1/4)[/tex].
Thus, the equation of the second line would be of the form [tex]\displaystyle y = (1/4)\, x + b_{2}[/tex] for some constant [tex]b_{2}[/tex] that needs to be found.
Make use of the fact that the second line goes through the point [tex](-4,\, 1)[/tex] to find the value of [tex]b_{2}[/tex]. Since the point [tex](-4,\, 1)\![/tex] is on the second line, the equation of the second line should hold after substituting in [tex]x = (-4)[/tex] and [tex]y = 1[/tex]. That is:
[tex]1 = (1/4)\times (-4) + b_{2}[/tex].
Solve this equation for [tex]b_{2}[/tex]:
[tex]b_{2} = 2[/tex].
Therefore, the equation of the second line would be [tex]y = (1/4)\, x + 2[/tex].
