Two proposed transit routes are modeled by the lines \( 3x - 4y = 12 \) and \( 2x + 5y = 10 \). Find their intersection point.

["Finding the Intersection Point of Two Proposed Transit Routes: Solving the System ( 3x - 4y = 12 ) and ( 2x + 5y = 10 )", "Urban planning and sustainable transit design rely heavily on modeling proposed routes mathematically. Two candidate transit lines are represented by the equations ( 3x - 4y = 12 ) and ( 2x + 5y = 10 ). Understanding where these routes intersect is crucial for optimizing transfers, minimizing delays, and improving connectivity in public transportation networks. In this article, we model these transit lines as linear equations, solve for their intersection point, and explain its significance in urban mobility.", "### Modeling Transit Lines Mathematically", "Transit routes are often represented as straight lines due to their simplicity and alignment with infrastructure. The given lines:", "- Line 1: ( 3x - 4y = 12 )\n- Line 2: ( 2x + 5y = 10 )", "model idealized transit paths, where ( x ) and ( y ) correspond to geographic coordinates on a city map. Finding their intersection determines whether the two proposed routes cross at a specific point—critical for determining stop placement or potential integration.", "### Solving the System of Equations", "To find the intersection, we solve the system algebraically using either substitution or elimination. Here, we use elimination for clarity.", "Step 1: Align coefficients for elimination\nWe aim to eliminate one variable. Let’s eliminate ( x ) by making coefficients matching. Multiply the first equation by 2 and the second by 3:", "[\n\begin{align}\n(3x - 4y &= 12) \ imes 2 \quad \Rightarrow \quad 6x - 8y = 24 \\n(2x + 5y &= 10) \ imes 3 \quad \Rightarrow \quad 6x + 15y = 30\n\end{align}\n]", "Step 2: Subtract equations to eliminate ( x )\nSubtract the first new equation from the second:", "[\n(6x + 15y) - (6x - 8y) = 30 - 24\n]\n[\n6x + 15y - 6x + 8y = 6 \quad \Rightarrow \quad 23y = 6\n]\n[\ny = \frac{6}{23}\n]", "Step 3: Substitute ( y = \frac{6}{23} ) into one original equation\nUse ( 3x - 4y = 12 ):", "[\n3x - 4\left(\frac{6}{23}\right) = 12\n]\n[\n3x - \frac{24}{23} = 12\n]\n[\n3x = 12 + \frac{24}{23} = \frac{276}{23} + \frac{24}{23} = \frac{300}{23}\n]\n[\nx = \frac{300}{23} \cdot \frac{1}{3} = \frac{100}{23}\n]", "### The Intersection Point", "The two transit lines intersect at the point:", "[\n\left( \frac{100}{23},\ \frac{6}{23} \right)\n]", "### Significance in Transit Planning", "This intersection point represents a strategic location where passengers from both proposed routes could transfer, enhancing multimodal connectivity. Urban planners use such intersections to evaluate accessibility, optimize stop placement, and ensure efficient route integration. Modeling these routes mathematically ensures data-driven decisions in city transit development.", "### Conclusion", "The model confirms that transit lines defined by ( 3x - 4y = 12 ) and ( 2x + 5y = 10 ) intersect at ( \left( \frac{100}{23},\ \frac{6}{23} \right) ), offering a concrete geographic reference for future transit integration. By leveraging algebraic techniques, planners can accurately design networks that meet growing urban mobility demands.", "---\nKeywords: transit route intersection, ( 3x - 4y = 12 ), ( 2x + 5y = 10 ), urban transit planning, solving linear systems, geographic coordinates for transit."]