English

Linear Functions (Applied)

🎯 Learning Goals

  • Solve real-world problems using linear functions
  • Understand the meaning of the intersection point of two linear graphs

💡 Why Learn This?

Comparing mobile phone plans, calculating when two moving objects will meet, or finding the break-even point in a business all require finding the intersection of linear functions. This is where math meets real-world decision-making.

The Intersection Point = The Break-Even Point

When you graph two linear functions, the point where they cross (intersect) is special. At this exact point, both the x and y values are the same for both situations. It tells you exactly when two different plans or scenarios become equal.

Real-world Scenario

  • Plan A: $10/month + $2/GB. Plan B: $20/month + $1/GB.
  • The graphs intersect at 10GB. Below 10GB, Plan A is cheaper. Above 10GB, Plan B is cheaper.

⚠️ Common Pitfalls

A common mistake is finding the intersection point but failing to interpret what it means. If x=10 and y=30 at the intersection, you must remember that x is GBs and y is dollars.

Phone Plan Comparator

Adjust the base fee and cost per GB for two plans to see where they intersect.

X (GB)
Y ($ / Yen)

Plan A (y = 3x + 10)

Base Fee ($):10
Rate/GB ($):3

Plan B (y = 1x + 30)

Base Fee ($):30
Rate/GB ($):1
Intersection:(x: 10.0, y: 40.0)

📝 Summary & Recap

  • The intersection of two linear graphs shows where both equations yield the same result.
  • You can find this point algebraically by setting the two equations equal to each other (e.g., 2x + 10 = x + 20).

Quick Drill

Test your understanding of linear intersections!

If Plan A is y = 20x and Plan B is y = 10x + 50, at what x do they cost the same?

🔍 Deep Dive (Optional)

In economics, the intersection of the 'Supply' and 'Demand' curves (which are often approximated as linear functions) determines the market price of a good. This is a real-world application of finding intersection points!