Answer:
[tex]\displaystyle y = \frac{1}{3}\, x + \frac{2}{3}[/tex].
Step-by-step explanation:
The goal is to find the equation of a line that goes through the given point [tex](1,\, 1)[/tex] and is parallel to line [tex]{\sf AB}[/tex].
For two lines to be parallel, their slopes must be equal. The slope of line [tex]{\sf AB}[/tex] is not given. However, since the coordinates of both [tex]{\sf A}[/tex] and [tex]{\sf B}[/tex] are available, the slope of the line between these two points can be found from this information.
In other words, the equation of the requested line can be found in the following steps:
The slope of a line between two points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] is:
[tex]\displaystyle (\text{slope}) = \frac{(\text{rise})}{(\text{run})} = \frac{y_{1} - y_{0}}{x_{1} - x_{0}}[/tex].
In this question, point [tex]{\sf A}[/tex] is at [tex](-2,\, 1)[/tex] while point [tex]{\sf B}[/tex] is at [tex](4,\, 3)[/tex]. The slope of the line between them would be:
[tex]\displaystyle m = \frac{3 - 1}{4 - (-2)} = \frac{1}{3}[/tex].
Since the requested line is parallel to line [tex]{\sf AB}[/tex], the slope of that line would also be [tex](1/3)[/tex].
If a line of slope [tex]m[/tex] goes through a point [tex](x_{0},\, y_{0})[/tex], the point-slope equation of that line would be:
[tex](y - y_{0}) = m\, (x - x_{0})[/tex].
For the requested line with slope [tex](1/3)[/tex] and goes through [tex](1,\, 1)[/tex], the point-slope equation would be:
[tex]\displaystyle (y - 1) = \frac{1}{3}\, (x - 1)[/tex].
Simplify to obtain:
[tex]\displaystyle y = \frac{1}{3}\, x + \frac{2}{3}[/tex].
