Section 5.5 - Parallel And Perpendicular Lines

Explanation
The equation of a line that is parallel to y = -3/5x + 6 will have the same slope as -3/5. Therefore, the correct answer is y = -3/5x + 3 because it has the same slope of -3/5 as the original equation.

Explanation
The equation y = -5/4x - 17 is the equation of a line perpendicular to y = 4/5x + 6 because the slopes of perpendicular lines are negative reciprocals of each other. The slope of y = 4/5x + 6 is 4/5, so the slope of the perpendicular line is -5/4. Additionally, the y-intercept of the perpendicular line is -17, which is different from the y-intercept of the original line. Therefore, y = -5/4x - 17 is the equation of a line perpendicular to y = 4/5x + 6.

Explanation
The line segments with endpoints at P(4, 5) and Q(-3, 2) have a slope of -1/7. In order for a line to be parallel to this line, it must have the same slope of -1/7. The line formed by the endpoints M (1, 2) and N (-6, -1) has a slope of -1/7, making it parallel to the given line segments. Therefore, M (1, 2) and N (-6, -1) form a line that is parallel to the line segments with endpoints at P(4, 5) and Q(-3, 2).

Explanation
The line that is perpendicular to the line segment AB(-2, -1) and B(6, -3) will have a slope that is the negative reciprocal of the slope of AB. The slope of AB can be calculated as (change in y / change in x) = (-3 - (-1)) / (6 - (-2)) = -2/4 = -1/2. The negative reciprocal of -1/2 is 2. Therefore, the line that is perpendicular to AB will have a slope of 2. Looking at the given endpoints, the line formed by C(8, 7) and D(9, 11) will have a slope of (11 - 7) / (9 - 8) = 4/1 = 4, which is the negative reciprocal of -1/2. Hence, C(8, 7) and D(9, 11) form a line that is perpendicular to AB.

Explanation
The equation y = 5 is the equation of a line that is parallel to the x-axis because it does not contain any term with x. This means that the value of x does not affect the value of y, and the line will be a horizontal line at y = 5. Additionally, the equation passes through the point (-2, 5) because when x = -2, y = 5. Therefore, the equation y = 5 satisfies both conditions of being parallel to the x-axis and passing through the point (-2, 5).

Explanation
The equation of a line that is parallel to the y-axis will have a constant x-value. Since the line passes through the point (7, -3), the x-value must be 7. Therefore, the correct answer is x = 7.

Explanation
The equation y = -3x - 7 is the equation of a line that is parallel to the line defined by y = -3x - 2. This is because the slope of both lines is -3, which means they have the same steepness. Additionally, the line y = -3x - 7 passes through the point (-2, -1), as required in the question. Therefore, this equation satisfies both conditions and is the correct answer.

Explanation
The equation of a line that is perpendicular to another line can be found by taking the negative reciprocal of the slope of the original line. The original 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 equation of a line perpendicular to -6x + 9y - 12 = 0 will have a slope that is 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 line defined by -8x + 2y - 6 = 0 has a y-intercept of 3. Therefore, the line perpendicular to -6x + 9y - 12 = 0 with the same y-intercept will have an equation of y = -3/2x + 3.

Explanation
The value of k in the graph is -3 because it is the only value that is shown in the given options.