Parallel & Perpendicular Lines

Explanation
The two lines are parallel because their slopes are equal. The slope of line A can be calculated as (5-9)/(6-0) = -4/6 = -2/3. The slope of line B can be calculated as (-7-(-3))/(0-(-6)) = -4/6 = -2/3. Since both slopes are equal, the lines are parallel.

Explanation
To determine if two lines are parallel, perpendicular, or neither, we need to compare their slopes. Line A has a slope of (6-1)/(-2-5) = -5/7. Line B has a slope of (-1-5.5)/(-1-5) = -6.5/-6 = 13/12. Since the slopes of the two lines are not equal, they are not parallel. Additionally, the product of their slopes (-5/7 * 13/12) is not equal to -1, so they are not perpendicular either. Therefore, the lines are neither parallel nor perpendicular.

Explanation
The equation y = -3/5x + 3 is the equation of a line parallel to y = -3/5x + 6 because it has the same slope (-3/5) as the given equation. The y-intercept (3) is different, but this does not affect the parallel nature of the lines.

Explanation
The equation of a line perpendicular to y = 4/5x + 6 would have a slope that is the negative reciprocal of 4/5. The negative reciprocal of 4/5 is -5/4. Therefore, the correct answer is y = -5/4x - 17.

Explanation
The equation y = 5 represents a line that is parallel to the x-axis because the y-coordinate remains constant at 5 regardless of the value of x. This line passes through the point (-2, 5) as the y-coordinate is indeed 5 when x = -2.

Explanation
The equation x = 7 represents a vertical line that is parallel to the y-axis. This is because the value of x remains constant at 7, while the value of y can vary. Therefore, any point on this line will have an x-coordinate of 7. Since the equation passes through the point (7, -3), it satisfies the condition and is the correct answer.

Explanation
The equation y = -3x - 7 is the equation of a line that is parallel to the line defined by y = -3x - 2 because it has the same slope (-3). Additionally, it passes through the point (-2, -1) which satisfies the equation.

Explanation
The equation of a line that is perpendicular to the line defined by 4x - 5y - 12 = 0 can be found by taking the negative reciprocal of the slope of the given line. The given line has a slope of 4/5, so the perpendicular line will have a slope of -5/4. The y-intercept remains the same, so the equation of the perpendicular line is y = -5/4x - 2.

Explanation
The given question asks for the equation of a line that is perpendicular to -6x + 9y - 12 = 0. To find the equation of a line perpendicular to a given line, we need to find the negative reciprocal of the slope of the given line. The given line has a slope of 6/9, which simplifies to 2/3. The negative reciprocal of 2/3 is -3/2.

The question also states that the line should have the same y-intercept as the line defined by -8x + 2y - 6 = 0. To find the y-intercept of this line, we set x = 0 and solve for y. By doing this, we find that the y-intercept is 3.

Therefore, the equation of the line perpendicular to -6x + 9y - 12 = 0 with the same y-intercept as -8x + 2y - 6 = 0 is y = -3/2x + 3.

Explanation
The value of k in the graph can be determined by looking at the position of the point on the y-axis. In this case, the point is located at -3 on the y-axis, so the value of k is -3.

Explanation
To determine whether lines A and B are parallel, perpendicular, or neither, we can examine the slopes of the lines.

For line A, the slope (m1) can be calculated as:

m1 = (y2 - y1) / (x2 - x1) = (7 - 3) / (-3 - (-8)) = 4 / 5

For line B, the slope (m2) can be calculated as:

m2 = (3 - 4) / (-3 - (-5)) = -1 / 2

Now, let's compare the slopes:

Lines A and B are neither parallel nor perpendicular because their slopes are not equal (not parallel) and the product of their slopes is not -1 (not perpendicular). They have different slopes, so they are neither parallel nor perpendicular to each other.