An entomologist is studying the trajectory of a butterfly modeled as a line passing through \( (2, -3) \) and \( (5, 6) \). Determine the \(y\)-intercept of this line.

["Title: How to Calculate the (y)-Intercept of a Line: A Butterfly Flight Path Example", "Meta Description:\nLearn how to find the (y)-intercept of a line using real-world data—like the flight path of a butterfly—through an example involving two points: ( (2, -3) ) and ( (5, 6) ).", "---", "Understanding the Path of a Butterfly: A Mathematical Approach", "Butterflies flutter with grace, but did you know their flight paths can be modeled mathematically? Just like a straight line on a graph, a butterfly’s movement through open air often follows a linear trajectory—especially when observed over short distances. Today, we explore a classic problem: determining the (y)-intercept of a line representing this flight path, given two specific points along the journey.", "### The Butterfly’s Path: Two Key Points", "An entomologist studying butterfly flight measures two key positions along its trajectory:", "- Point (A = (2, -3)): represented by coordinates ( (x = 2, y = -3) )\n- Point (B = (5, 6)): where ( (x = 5, y = 6) )", "These points describe where the butterfly was observed at two instants, allowing scientists to model its path as a straight line.", "### Step 1: Find the Slope of the Line", "The equation of a line is ( y = mx + b ), where (m) is the slope. First, compute (m) using the formula:", "[\nm = \frac{y_2 - y_1}{x_2 - x_1}\n]", "Substituting ( (x_1, y_1) = (2, -3) ) and ( (x_2, y_2) = (5, 6) ):", "[\nm = \frac{6 - (-3)}{5 - 2} = \frac{9}{3} = 3\n]", "So, the slope (m = 3).", "### Step 2: Use Point-Slope Form to Find the (y)-Intercept", "Now that we know (m = 3), we use the point-slope form of a line:", "[\ny - y_1 = m(x - x_1)\n]", "Using point (A = (2, -3)):", "[\ny + 3 = 3(x - 2)\n]", "Expand and simplify:", "[\ny + 3 = 3x - 6\n]\n[\ny = 3x - 6 - 3\n]\n[\ny = 3x - 9\n]", "### Final Step: Identify the (y)-Intercept", "In the slope-intercept form ( y = mx + b ), (b) is the (y)-intercept—the value of (y) when (x = 0).", "Here, ( b = -9 ).", "---", "Conclusion: The Butterfly’s Starting Point on the Graph", "The (y)-intercept of the butterfly’s flight path line is (-9). This means, in this model, if the butterfly’s trajectory were extended straight down from the point ( (2, -3) ), it would cross the (y)-axis at ( y = -9 ).", "Understanding the (y)-intercept helps entomologists predict future positions, analyze flight patterns, and communicate results clearly in ecological studies.", "---", "Keywords: (y)-intercept, line equation, slope calculation, butterfly flight path, entomology data analysis, slope-intercept form, math modeling, trajectory analysis.", "---", "Further Reading:\nExplore how linear equations model animal movement, or learn how scientists use coordinate geometry in wildlife tracking."]