Section 5.3 - Slope Of A Line

Explanation
The slope of a line can be calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates of point M are (4, -7) and the coordinates of point N are (2, -5). Plugging these values into the formula, we get (-5 - (-7)) / (2 - 4) = (-5 + 7) / (2 - 4) = 2 / -2 = -1. Therefore, the slope of the line through points M and N is -1.

Explanation
To calculate the slope of a line passing through two points, we use the formula: slope = (change in y) / (change in x). In this case, the change in y is -2 - 0 = -2, and the change in x is 4 - (-6) = 4 + 6 = 10. Therefore, the slope is -2/10 = -1/5.

Explanation
The point (-6, 0) would be on the line passing through point D(-1, -3) with a slope of -3. This can be determined by using the point-slope formula, which states that the equation of a line passing through point (x₁, y₁) with a slope m is given by y - y₁ = m(x - x₁). Plugging in the values for D(-1, -3) and the slope of -3, we get y - (-3) = -3(x - (-1)), which simplifies to y + 3 = -3x - 3. By solving for y, we get y = -3x - 6. Substituting x = -6 into the equation, we find that y = 0, confirming that (-6, 0) is on the line.

Explanation
The line passing through point R(5, -2) with a slope of 1/2 can be represented by the equation y = (1/2)x - 9/2. By substituting x = 7 into this equation, we can find the y-coordinate of the point on the line: y = (1/2)(7) - 9/2 = -1. Therefore, the coordinates (7, -1) would be the coordinates of one other point on the line.

Explanation
The points (-1, -5), (1, 1), and (0, -2) are collinear because they lie on the same straight line. The slope of the line passing through (-1, -5) and (1, 1) is (1-(-5))/(1-(-1)) = 6/2 = 3. The slope of the line passing through (-1, -5) and (0, -2) is (-2-(-5))/(0-(-1)) = 3/1 = 3. Since both slopes are equal, the points (-1, -5), (1, 1), and (0, -2) are collinear.

Explanation
The point (2, 6) is collinear with (-4, 2) and (5, 8) because it lies on the same line as these two points. Collinear points are points that lie on the same straight line. In this case, the slope between (-4, 2) and (5, 8) is the same as the slope between (-4, 2) and (2, 6), indicating that all three points lie on the same line. Therefore, (2, 6) is collinear with (-4, 2) and (5, 8).

Explanation
The Quicks are driving at an average speed of 67 km/hr. This can be calculated by finding the average speed between the two distances and the corresponding driving times. The difference in distance is 250km - 150km = 100km. The difference in driving time is 4.75 hours - 3.25 hours = 1.5 hours. Therefore, the average speed is 100km / 1.5 hours = 66.67 km/hr, which can be rounded to 67 km/hr.

Explanation
To find the average rate of speed, we need to calculate the total distance covered and divide it by the total time taken. Katie ran 5 km in 30 minutes and 34 km in 2.5 hours, which is equivalent to 150 minutes. Therefore, the total distance covered is 5 km + 34 km = 39 km, and the total time taken is 30 minutes + 150 minutes = 180 minutes. Dividing the total distance by the total time gives us the average rate of speed, which is 39 km / 180 minutes = 0.2167 km/minute. Converting this to km/h, we get 0.2167 km/minute * 60 minutes/hour = 13 km/h. Therefore, the correct answer is 13.75 km/h.

Explanation
To find the value of n, we can use the slope formula: slope = (y2 - y1) / (x2 - x1). Given that the slope is 1 and the coordinates of F are (4, 3), we can substitute these values into the formula: 1 = (-1 - 3) / (n - 4). Simplifying this equation, we get -4 = -4 / (n - 4). Multiplying both sides by (n - 4), we have -4(n - 4) = -4. Expanding the equation, we get -4n + 16 = -4. Rearranging the equation, we find that -4n = -20. Dividing both sides by -4, we get n = 5. Therefore, the value of n that satisfies the given conditions is 5.