Question: Find the intersection point of the lines $ 3x - 4y = 12 $ and $ 5x + 2y = 10 $. - Treasure Valley Movers

Find the intersection point of the lines $ 3x - 4y = 12 $ and $ 5x + 2y = 10 $ — And How It Matters

Imagine walking through a city where two paths cross — not just physically, but conceptually. In math, geometry, and planning, an intersection is the key to understanding connections. Now, for curious minds exploring data, urban design, or real-world applications, finding this intersection unlocks deeper insight into problem-solving, design logic, and trend analysis. That intersection—where two linear equations meet—doesn’t just solve a puzzle. It shapes strategies, decisions, and even digital discovery. Discover the math behind the crossroads.

Why People Are Talking About This Question in the US Market Now

In recent years, interest in spatial relationships and data visualization has surged across academic, urban planning, and tech communities. As cities grow and digital tools become more essential, understanding how systems intersect — like transport routes, economic zones, or infrastructure networks — has never been more critical. This question surfaces frequently in discussions about smart city planning, logistics efficiency, real estate development, and public policy. Users visiting content platforms like Discover are increasingly seeking clear, accurate steps to unpack such intersections, blending curiosity with practical intent. The question reflects a broader national interest in problem-solving through tangible, data-driven insight.

How to Find the Intersection of $ 3x - 4y = 12 $ and $ 5x + 2y = 10 $ — Step by Step

To find the intersection point, we solve the two equations simultaneously. Start with:
$ 3x - 4y = 12 $  (1)
$ 5x + 2y = 10 $  (2)

First, eliminate one variable. Multiply equation (2) by 2 to align the $ y $-coefficients:
$ 2(5x + 2y) = 2(10) $ → $ 10x + 4y = 20 $ (3)

Now add equation (1) and (3):
$ (3x - 4y) + (10x + 4y) = 12 + 20 $
$ 13x = 32 $ → $ x = \frac{32}{13} $

Next, substitute $ x = \frac{32}{13} $ into a simplified version of equation (2). Divide equation (2) by 2:
$ \frac{5}{2}x + y = 5 $ → $ y = 5 - \frac{5}{2}x $

Plug in $ x $:
$ y = 5 - \frac{5}{2} \cdot \frac{32}{13} = 5 - \frac{160}{26} = 5 - \frac{80}{13} = \frac{65 - 80}{13} = \frac{-15}{13} $

Thus, the intersection point is $ \left( \frac{32}{13}, -\frac{15}{13} \right) $. This precise solution helps visualize how mathematical logic translates into real-world applications across fields like infrastructure mapping, economic