Parallel and Perpendicular Lines Equations
Find the equation of the line parallel to the line shown in the graph passing through the point (-2, 3).
A y = 2/3x - 13/3
B y = 3/2x + 13/3
D y = 2/3x + 13/3
Find the equation of the line perpendicular to the line shown in the graph passing through the point (-2, 3).
A y = -3/2x
B y = 3/2x
C y = 3/2x-3
D y = -3/2x + 3
To find the equation of the line parallel to the line shown in the graph, we first need to find the slope of the given line. Since we don't have the equation of the given line, let's assume its equation is y = mx + b, where m is the slope and b is the y-intercept.
Since the line is parallel to the given line, it must have the same slope. So, the equation of the parallel line is y = mx + c, where c is the y-intercept of the parallel line.
Now, we are given that the parallel line passes through the point (-2, 3). So, substituting this point into the equation of the parallel line, we get:
3 = m(-2) + c
We only have options with slopes of 2/3 and 3/2. Thus, we must test these to see which is the correct slope:
1) If m = 2/3, then the equation is y = (2/3)x + c, substituting the point (-2, 3) we get:
3 = (2/3)(-2) + c => c = 13/3
Thus, the correct equation for the parallel line is y = (2/3)x + 13/3, which corresponds to answer D.
Now, let's find the equation of the line perpendicular to the given line. Perpendicular lines have slopes that are negative reciprocals of each other. So, if m is the slope of the given line, then the slope of the perpendicular line is -1/m.
Again, we have slopes of 2/3 and 3/2 as our options. We must find out which is the correct slope:
1) If m = 2/3, then the perpendicular slope is -1/(2/3) = -3/2.
2) If m = 3/2, then the perpendicular slope is -1/(3/2) = -2/3.
Since we are now looking for an equation in the form y = mx + c with a slope of either -3/2 or -2/3, and it goes through the point (-2, 3). We must try both slopes:
1) If the slope is -3/2, then the equation is y = (-3/2)x + c, substituting the point (-2, 3) we get:
3 = (-3/2)(-2) + c => c = 3 - 3 = 0
Thus, the correct equation for the perpendicular line is y = (-3/2)x + 0, that is, y = (-3/2)x, which corresponds to answer A.
To determine the equation of a line parallel or perpendicular to a given line, you need to understand the relationship between the slopes of the lines.
For two lines to be parallel, they must have the same slope. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope. So, to find a line parallel to a given line, you need to find a line with the same slope.
In this case, the given line has the equation y = 2/3x - 13/3. You can see that the slope of this line is 2/3. To find a line parallel to this, you need to use the same slope.
Therefore, the equation for a line parallel to the given line is y = 2/3x + b. To find the value of b, you can substitute the coordinates (-2, 3) of the point through which the line passes into the equation and solve for b.
Plugging in the values, you get 3 = (2/3)(-2) + b. Solving for b, you find b = 13/3.
So, the equation of the line parallel to the given line and passing through the point (-2, 3) is y = 2/3x + 13/3.
Now, let's move on to finding the equation of a line perpendicular to the given line. For two lines to be perpendicular, the product of their slopes should be -1.
The given line has a slope of 2/3. To find a line perpendicular to this, you need to find a line with a slope that, when multiplied by 2/3, gives -1.
The slope that meets this requirement is -3/2. So, the equation for a line perpendicular to the given line is y = -3/2x + b.
Again, to find the value of b, substitute the coordinates (-2, 3) into the equation and solve for b.
Plugging in the values, you get 3 = (-3/2)(-2) + b. Solving for b, you find b = 3.
Therefore, the equation of the line perpendicular to the given line and passing through the point (-2, 3) is y = -3/2x + 3.
So, the final answers are:
- The equation of the line parallel to the given line passing through the point (-2, 3) is y = 2/3x + 13/3 (choice D).
- The equation of the line perpendicular to the given line passing through the point (-2, 3) is y = -3/2x + 3 (choice D).
To find the equation of the line parallel to the given line and passing through the point (-2, 3):
First, determine the slope of the given line. The given equation is in the form y = mx + b, where m represents the slope.
Comparing the given equation "y = (2/3)x - (13/3)" with the standard form "y = mx + b," we find that the slope of the given line is 2/3.
Now that we have the slope (2/3) and the point (-2, 3), we can use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1),
where (x1, y1) represents the coordinates of the point on the line (in this case, (-2, 3)) and m represents the slope of the parallel line (in this case, 2/3).
Plugging in the values gives:
y - 3 = (2/3)(x - (-2)).
Simplifying, we have:
y - 3 = (2/3)(x + 2).
Expanding and rearranging:
y - 3 = (2/3)x + 4/3.
Adding 3 to both sides:
y = (2/3)x + 4/3 + 9/3.
Simplifying further:
y = (2/3)x + 13/3.
Therefore, the equation of the line parallel to the given line and passing through the point (-2, 3) is y = (2/3)x + 13/3.
Now, to find the equation of the line perpendicular to the given line and passing through the point (-2, 3):
Using the same method as before, we can determine the slope of the given line. In this case, the slope is 2/3.
The slope of a line perpendicular to another line can be found by taking the negative reciprocal of the slope of the given line.
So, the slope of the perpendicular line is -3/2.
Now, we can use the point-slope form of a linear equation with the slope -3/2 and the point (-2, 3):
y - 3 = (-3/2)(x - (-2)).
Simplifying, we get:
y - 3 = (-3/2)(x + 2).
Expanding and rearranging:
y - 3 = (-3/2)x - 3.
Adding 3 to both sides:
y = (-3/2)x + 3 - 3.
Simplifying further:
y = (-3/2)x.
Therefore, the equation of the line perpendicular to the given line and passing through the point (-2, 3) is y = (-3/2)x.