Answer:
a. The slope of a line parallel to the given line is 1
b. A point on the line parallel to the given line, passing through (−4, 2), is (1,7)
c. The slope of the line perpendicular to the given line is -1
d. A point on the line perpendicular to the given line, passing through (−4, 2), is (3,-5)
Step-by-step explanation:
The equation of the line in Slope-intercept form is:
[tex]y=mx+b[/tex]
Where m is the slope and b is the y-intercept.
a. For the line [tex]y = x - 4[/tex]
You can identify that:
[tex]m=1[/tex]
By definition, two lines are parallel if they have the same slope. Then, the slope of a line parallel to the given line is:
[tex]m=1[/tex]
b. The equation of the line in Point-slope form is:
[tex]y -y_1 = m(x - x_1)[/tex]
Where m is the slope and ([tex]x_1,y_1[/tex]) is a point of the line.
Given the point (-4,2), substitute this point and the slope of the line into the equation:
[tex]y -2 = (x +4)[/tex]
Give a value to "x", substitute it into this equation and solve for "y":
For [tex]x=1[/tex] :
[tex]y -2 = (1 +4)[/tex]
[tex]y= 5+2[/tex]
[tex]y= 7[/tex]
Then, you get the point (1,7)
c. The slopes of perpendicular lines are negative reciprocals, then the slope of a line perpendicular to the given line is:
[tex]m=-\frac{1}{1}\\\\m=-1[/tex]
d. Given the point (-4,2), substitute this point and the slope of the line into the equation:
[tex]y -2 = -1(x +4)[/tex]
[tex]y -2 = -(x +4)[/tex]
Give a value to "x", substitute it into this equation and solve for "y":
For [tex]x=3[/tex] :
[tex]y -2 = -(3 +4)[/tex]
[tex]y= -7+2[/tex]
[tex]y= -5[/tex]
Then, you get the point (3,-5)
