A line passes through the points (2, 5) and (6, 17). What is the slope of the line?
["Understanding the Slope of a Line: The Case of Points (2, 5) and (6, 17)", "When studying algebra and geometry, one of the key concepts you’ll encounter is the slope of a line. The slope represents how steep a line is and indicates the rate of change between two points on the line. But how do you calculate the slope using two specific points? Let’s explore this using the well-known points: (2, 5) and (6, 17).", "### What Is Slope?", "In coordinate geometry, the slope between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the formula:", "[\nm = \frac{y_2 - y_1}{x_2 - x_1}\n]", "This formula computes the ratio of the vertical change (rise) to the horizontal change (run) between the points. A positive slope means the line rises as it moves left to right; a negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope corresponds to a vertical line.", "### Applying the Formula", "Given the points:", "- Point 1: ((x_1, y_1) = (2, 5))\n- Point 2: ((x_2, y_2) = (6, 17))", "Plug these values into the slope formula:", "[\nm = \frac{17 - 5}{6 - 2} = \frac{12}{4} = 3\n]", "### Interpreting the Result", "The slope ( m = 3 ) means that for every 1 unit increase in the (x)-coordinate, the (y)-coordinate increases by 3 units. This steep positive slope confirms that the line rises steeply from left to right.", "### Visualizing the Line", "If you were to plot the points:", "- From (2, 5), moving 4 units right (to x = 6) leads to 12 units up (from y = 5 to y = 17)\n- This matches the slope calculation of ( \frac{12}{4} = 3 )", "### Why Slope Matters", "Knowing the slope helps predict values, graph lines accurately, and understand relationships in real-life scenarios—from calculating speed to analyzing financial trends.", "### Summary", "To find the slope of a line through the points (2, 5) and (6, 17), subtract the y-coordinates and divide by the x-coordinates:", "[\nm = \frac{17 - 5}{6 - 2} = \frac{12}{4} = 3\n]", "The slope of the line is 3.", "Understanding this fundamental concept is essential for mastering linear equations and their applications in mathematics and beyond."]
