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.
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
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