Find (a) the equation of a line that is parallel to the given line and includes the given point, and (b) the equation of a line that is perpendicular to the given line through the given point. Write both answers in slope-intercept form. (c) Graph both of these lines on the same axes and submit your graph. Please show all of your work.
y = -2x + 3, (-2, -3)
Just use the point-slope form and then rearrange terms to slope-intercept form:
The line through (-2,-3) with slope -2:
(y-(-3))/(x-(-2)) = (y+3)/(x+2) = -2
The other line has slope -1/(-2) = 1/2.
Go for it.
To find the equation of a line that is parallel to the given line and includes the given point, we need to know that parallel lines have the same slope. The given line has a slope of -2. So, the parallel line will also have a slope of -2.
(a) To find the equation of the parallel line, we can use the point-slope form of a linear equation, which is:
y - y1 = m(x - x1),
where m is the slope and (x1, y1) is the given point.
Let's substitute the known values:
y - (-3) = -2(x - (-2)),
Simplifying further:
y + 3 = -2(x + 2).
Distribute the -2:
y + 3 = -2x - 4.
Now, isolate y:
y = -2x - 4 - 3.
Simplify:
y = -2x - 7.
So, the equation of the line that is parallel to the given line and includes the given point (-2, -3) is y = -2x - 7.
(b) To find the equation of a line that is perpendicular to the given line and passes through the given point, we need to know that perpendicular lines have negative reciprocal slopes. The given line has a slope of -2, so the perpendicular line will have a slope of 1/2 (-1/(-2)).
Now, we can use the same process as before using the point-slope form of a linear equation:
y - (-3) = 1/2(x - (-2)),
Simplifying further:
y + 3 = 1/2(x + 2).
Distribute the 1/2:
y + 3 = 1/2x + 1.
Isolate y:
y = 1/2x + 1 - 3.
Simplify:
y = 1/2x - 2.
So, the equation of the line that is perpendicular to the given line and passes through the given point (-2, -3) is y = 1/2x - 2.
(c) Now, let's graph both lines on the same axes:
To graph y = -2x + 3 (the given line), we can start by plotting the y-intercept, which is the point (0, 3). Then, since the slope is -2, we can use the slope to find other points on the line. For example, when x = 1, y = -2(1) + 3 = 1, so the point (1, 1) is on the line. Connect these points to get a line.
To graph y = -2x - 7 (the parallel line), we can use the same process. Using the y-intercept, we have the point (0, -7). Then, as the slope is -2, we can find other points on the line. For example, when x = 1, y = -2(1) - 7 = -9, giving us the point (1, -9). Connect these points to get a line.
To graph y = 1/2x - 2 (the perpendicular line), start by plotting the y-intercept, which is the point (0, -2). Then, as the slope is 1/2, we can find other points on the line. For example, when x = 2, y = 1/2(2) - 2 = -1, so we have the point (2, -1). Connect these points to get a line.
Once you have graphed all three lines, you can submit your graph.