Given the relationship of the two lines....please help help me, been on this for 2 hours.

Question

Given the description of lines A, B, and C, choose the relationship that best describes the relationship between two of the lines. 

Line A: is a line that goes through point (-1,3) and is parallel to y=2x+3

Line B: is a line that goes through point (-2,2) and is perpendicular to y=-2x-6

Line C: is a line that goes through the origin and is parallel to y=-1/2x+2

https://docs.google.com/document/d/11aQOictwevp2FAk1H8ZKuuSn25lZ4PKPWT5N2cMn4Hs/edit (link to image that comes with problem) 

F. Lines A and B are perpendicular 

G. Lines A and B are parallel 

H. Lines A and C are perpendicular 

J. Lines A and C are parallel

Answer 1

The key to this problem is to recognize that parallel lines have the same slope, while lines that are perpendicular have slopes that are negative reciprocals to one another. Since the equations for all three lines are already in standard form (y = mx + b, where m is the coefficient of x and b is the y-intercept), all we have to do is to look at m, which represents the slope of the line.

Since the coefficient of x for the line y = 2x + 3 is 2, the slope for this line is 2. Since parallel lines share the same slope, any line parallel to y = 2x + 3 will also have a slope of 2, including Line A. Likewise, Line C, being parallel to y = -1/2x + 2, will have a slope of -1/2, since this is the coefficient of x. The slopes 2 and -1/2 are negative reciprocals, so any two lines with these slopes will be perpendicular to one another. Therefore the answer is Choice H (Lines A and C are perpendicular).

Answer 2

First we need to find the equations of lines A, B, and C. To do so we need to remember that lines that are parallel have the same slope as each other and lines that are perpendicular to each other have opposite slopes. We also need to know the equation for a linear function, y = mx + b.

Line A: is a line that goes through point (-1, 3) and is parallel to y = 2x + 3.

Because line A is parallel to y = 2x + 3, their slopes must be equal (m = 2). Since the point of the line is given we can then solve for b by plugging in x and y.

y = 2x + b Given.

3 = 2(-1) + b Plug in x and y values.

3 = -2 + b Simplify.

5 = b Add 2 to both sides.

Therefore, our equation for line A is y = 2x + 5

Line B: is a line that goes through point (-2, 2) and is perpendicular to y = -2x - 6.

Because line B is perpendicular to y = -2x - 6, their slopes must be opposite (m = 1/2). Since the point of the line is given we can plug in x and y and solve for b.

y = (1/2)x + b Given.

2 = (1/2)(-2) + b Plug in x and y values.

2 = -1 + b Simplify.

3 = b Add 1 to both sides.

Therefore, our equation for line B is y = (1/2)x + 3

Line C: is a line that goes through the origin and is parallel to y = (-1/2)x + 2.

We will follow the same steps as Line A. Because line C is parallel it has the same slope (m = -1/2). The point of the line is given as the origin, (0,0). Plug in x and y values to get b.

y = (-1/2)x + b Given.

0 = (-1/2)(0) + b Plug in x and y values.

0 = 0 + b Simplify.

0 = b Simplify.

Therefore the equation for line C is y = (-1/2)x

Choose the relationship that best describes the relationship between two of the lines.

To find the relationship between these lines they must be graphed. You can do so on paper or in a graphing calculator like the one provided on desmos.com. When comparing lines A, B, and C, you will find that option H, lines A and C are perpendicular, best describes the relationship.