#### 1171.659**Question:** An oceanographer is using an autonomous underwater vehicle to track the movement of two distinct marine species. The vehicle records the path of each species as a linear equation. The first species follows the path given by the equation \( y = 2x + 3 \) and the second species follows \( y = -x + 7 \). Determine the coordinates where the paths of these two species intersect.

["Intersection of Marine Species Paths: A Detailed Breakdown Using Linear Equations", "In marine biology research, understanding how distinct species occupy and move through their underwater habitats is vital. Tracking the paths of marine organisms often involves mapping their movement lines—sometimes modeled as straight lines in coordinate space. In this article, we explore a practical application involving two species tracked by an autonomous underwater vehicle (AUV), whose movement is described by two linear equations. Today, we determine the precise coordinates where these paths intersect—an essential intersection point for ecological analysis.", "### The Problem: Finding the Intersection of Two Marine Trajectories", "An oceanographer uses an autonomous underwater vehicle to monitor two distinct marine species. The first species follows a linear path represented by the equation:", "[\ny = 2x + 3\n]", "This slope of 2 indicates a steep, upward trajectory across the sea floor or water column. The second species moves along a different route modeled by:", "[\ny = -x + 7\n]", "With a slope of -1, this path slopes downward, suggesting contrasting behavioral patterns like foraging or migratory tendencies.", "To uncover critical behavioral intersections—such as potential species interactions or feeding hotspots—the oceanographer must determine where these two paths cross.", "---", "### Step-by-Step Solution: Solving the System of Equations", "The intersection point of two linear equations corresponds to the unique ((x, y)) value satisfying both equations simultaneously. We solve the system:", "1. ( y = 2x + 3 )\n2. ( y = -x + 7 )", "Step 1: Set the equations equal\nSince both expressions equal ( y ), equating them yields:", "[\n2x + 3 = -x + 7\n]", "Step 2: Solve for ( x )\nAdd ( x ) to both sides:", "[\n2x + x + 3 = 7 \quad \Rightarrow \quad 3x + 3 = 7\n]", "Subtract 3 from both sides:", "[\n3x = 4\n]", "Divide by 3:", "[\nx = \frac{4}{3}\n]", "Step 3: Substitute ( x ) into one equation to find ( y )\nUsing the first equation ( y = 2x + 3 ):", "[\ny = 2\left(\frac{4}{3}\right) + 3 = \frac{8}{3} + 3 = \frac{8}{3} + \frac{9}{3} = \frac{17}{3}\n]", "---", "### Intersection Coordinates", "The paths of the two marine species intersect at:", "[\n\left( \frac{4}{3},\ \frac{17}{3} \right)\n]", "---", "### Significance for Marine Ecology", "This intersection point—where the trajectories of two marine species cross—offers valuable ecological insights. It may indicate a shared habitat zone, suggesting potential for species interactions such as predation, competition, or cooperative behavior. Understanding such spatial overlaps helps oceanographers predict biodiversity dynamics and optimize conservation strategies.", "---", "### Conclusion", "Using linear equations to model species movement enables precise tracking and intersection analysis. The computed intersection at ( \left( \frac{4}{3},\ \frac{17}{3} \right) ) embodies a mathematical convergence that mirrors ecological complexity beneath the waves. For researchers deploying autonomous vehicles, solving these paths not only advances science but enhances our stewardship of the ocean’s living systems.", "---", "Keywords: oceanography, underwater vehicle, autonomous underwater vehicle, marine species tracking, linear equations, path intersection, species interaction, ecological modeling, ( y = 2x + 3 ), ( y = -x + 7 ), coordinate geometry."]