Answer:
The equation of the line passes through (3, -2) and parallel to the given line:
The equation of the line passes through (3, -2) and perpendicular to the given line:
- [tex]y=-\frac{1}{3}x-1[/tex]
Step-by-step explanation:
Given the points on the graph line
(2, -1)
(1, -4)
Finding the slope between (2, -1) and (1, -4)
[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\left(x_1,\:y_1\right)=\left(2,\:-1\right),\:\left(x_2,\:y_2\right)=\left(1,\:-4\right)[/tex]
[tex]m=\frac{-4-\left(-1\right)}{1-2}[/tex]
[tex]m=3[/tex]
Equation of the line passes through (3, -2) and parallel to the given line.
We know that parallel lines have the same slope.
so the equation of the line parallel to the given line = 3
Thus, using the point-slope form of the line equation
[tex]y-y_1=m\left(x-x_1\right)[/tex]
where m is the slope of the line and (x₁, y₁) is the point
substituting the values m = 3 and the point (3, -2)
[tex]y - (-2) = 3(x-3)[/tex]
[tex]y+2 = 3x-9[/tex]
[tex]y = 3x-9-2[/tex]
[tex]y = 3x-11[/tex]
Therefore, the equation of the line passes through (3, -2) and parallel to the given line:
[tex]y = 3x-11[/tex]
The equation of the line passes through (3, -2) and perpendicular to the given line
We know that a line perpendicular to another line contains a slope that is the negative reciprocal of the slope of the other line, such as:
slope = m = 3
perpendicular slope = – 1/m = -1/3 = -1/3
Therefore, substituting the values of perpendicular slope = -1/3 and the point (3, -2) in the point-slope form of the line equation
[tex]y-\left(-2\right)=-\frac{1}{3}\left(x-3\right)[/tex]
[tex]y+2=-\frac{1}{3}\left(x-3\right)[/tex]
subtract 2 from both sides
[tex]y+2-2=-\frac{1}{3}\left(x-3\right)-2[/tex]
[tex]y=-\frac{1}{3}x-1[/tex]
Therefore, the equation of the line passes through (3, -2) and perpendicular to the given line:
[tex]y=-\frac{1}{3}x-1[/tex]
Conclusion:
The equation of the line passes through (3, -2) and parallel to the given line:
The equation of the line passes through (3, -2) and perpendicular to the given line:
- [tex]y=-\frac{1}{3}x-1[/tex]
