Answer:
[tex](0,\, 4)[/tex] would be the [tex]y[/tex]-intercept of this line.
Step-by-step explanation:
In this question, it is given that:
- The points [tex](-1,\, -3)[/tex] and [tex](2,\, 3)[/tex] are on line [tex]{\rm DF}[/tex].
- An unknown line is parallel to line [tex]{\rm DF}[/tex], and the point [tex]{\rm G}[/tex] is on that line.
The goal is to find the point where the unknown line intersects the [tex]y[/tex]-axis (the [tex]y[/tex]-intercept of this line.) This value can be obtained if the equation of this unknown line can be found.
Since the unknown line is parallel to line [tex]{\rm DF}[/tex], the slope of the two lines should be the same. Hence, one possible way to find the equation of the unknown line is to find the slope of line [tex]{\rm DF}[/tex], which would be the same as that of the unknown line. After that, obtain the equation of this unknown line in point-slope given the coordinates of point [tex]{\rm G}[/tex].
Overall, approach this question in the following steps:
- Find the slope of the line [tex]{\rm DF}[/tex] given the two points on that line.
- Find the equation of the unknown line in point-slope form given the slope of that line and the coordinates of point [tex]{\rm G}[/tex].
- Find the [tex]y[/tex]-intercept of the unknown line, which is the point on that line where the [tex]x[/tex]-coordinate is [tex]0[/tex].
In a cartesian plane, if two points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] (where [tex]x_{0} \ne x_{1}[/tex]) are on a line, the slope [tex]m[/tex] of that line would be:
[tex]\displaystyle m = \frac{y_{1} - y_{0}}{x_{1} - x_{0}}[/tex].
Hence, the slope of line [tex]{\rm DF}[/tex] would be:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}} = \frac{3 - (-3)}{2 - (-1)} = 2\end{aligned}[/tex].
Because the unknown line is parallel to line [tex]{\rm DF}[/tex], the slope of the unknown line should also be [tex]2[/tex].
In a cartesian plane, if a point [tex](x_{0},\, y_{0})[/tex] is on a line of slope [tex]m[/tex], the point-slope equation of that line would be:
[tex](y - y_{0}) = m\, (x - x_{0})[/tex].
Hence, the equation of the unknown line would be:
[tex]y = 2\, x + 4[/tex].
The [tex]y[/tex]-intercept of a line is the point where that line intersects the [tex]y[/tex]-axis. The [tex]x[/tex]-coordinate of a [tex]y[/tex]-intercept should always be [tex]0[/tex]. To find the [tex]y[/tex]-coordinate, substitute [tex]x = 0[/tex] into the equation of the line and solve for the value of [tex]y[/tex]. Hence, [tex](0,\, 4)[/tex] would be the [tex]y[/tex]-intercept of the unknown line in this question.
