Answer:
[tex](9, 8), \\(0, -7), \\(6, 3), \\[/tex]
Step-by-step explanation:
There is! Algebraically, we can find the equation of the line. Any points that the line passes through should make the equation of the line true.
In slope-intercept form, the equation of a line is given by [tex]y=mx+b[/tex], where [tex]m[/tex] is the slope of the line, [tex]b[/tex] is the y-intercept, and [tex](x,y)[/tex] represents any point the line passes through.
The slope of a line is given by the change in y-values over the change in x-values of two points the line passes through. We're given that the line passes through the points (-6, -4) and (-3, 1).
Let:
[tex](x_1, y_1)\implies (-6, -4)\\(x_2,y_2)\implies (-3, 1)[/tex] (doesn't matter which point you assign to which)
The slope of the line that passes through these points is [tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{1-(-4)}{-3-(-6)}=\frac{5}{3}[/tex]
Now substitute [tex]m=\frac{5}{3}[/tex] and any point (x, y) that the line passes through to solve for [tex]b[/tex] in [tex]y=mx+b[/tex]:
Using (-3, 1):
[tex]1=\frac{5}{3}(-3)+b, \\\\\frac{3}{3}=-\frac{15}{3}+b,\\\\b=\frac{3}{3}+\frac{15}{3}=\frac{18}{3}=6[/tex]
Thus, the equation of the line is [tex]y=\frac{5}{3}x+6[/tex].
I just realized that it's asking for the points that the line parallel to this line and passing through point [tex]p[/tex] goes through, but I'll keep all the information above as it may be useful.
Parallel lines always have equal slopes, by definition. Therefore, the slope of the line that passes through point [tex]p[/tex] and is parallel to the line above also has a slope of [tex]5/3[/tex]. Point [tex]p[/tex] is located at (3, -2). Thus, to find [tex]b[/tex], plug in [tex]m=5/3[/tex] and [tex]x=3, y=-2[/tex] into [tex]y=mx+b[/tex]:
[tex]-2=\frac{5}{3}(3)+b, \\-2=5+b, \\b=-2-5=-7[/tex]
Therefore, the equation of this line must be [tex]y=\frac{5}{3}x-7[/tex].
You can algebraically determine if a certain point passes through this line by seeing if the point makes this equation true. From the answer choices, the following points are passed through by this line:
[tex](9, 8)\:\checkmark, \\(0, -7)\:\checkmark, \\(6, 3)\:\checkmark, \\[/tex]
