Lesson 7.9

Special Systems

Sometimes all the variables disappear! When this happens, don't panic. It means the lines are either parallel (never cross) or identical (always cross).

Introduction

We learned in Lesson 2.11 that equations can have No Solution or Infinite Solutions. The same is true for systems. We need to recognize the "False Statement" and the "True Statement".

Past Knowledge

Lesson 2.11 (Special Solutions). is good. is bad.

Today's Goal

Interpret simple True/False statements as geometric facts about lines.

Future Success

This logic (contradiction vs. identity) is arguably more important in logic/philosophy than the math itself.

Key Concepts

The Two Strange Endings

The Lie

0 = 7

No Solution

The variables vanish and leave a lie. Parallel lines never touch.

The Truth

0 = 0

Infinite Solutions

The variables vanish and leave a truth. They are the same line.

Worked Examples

Example 1: The Lie (Parallel)

Basic

Solve the system.

Step 1: Subtract

Conclusion: No Solution

That is impossible. The lines are parallel.

Example 2: The Truth (Infinite)

Intermediate

Solve the system.

Step 1: Scale Up

Multiply top by :

Step 2: Add

Conclusion: Infinite Solutions

They are the exact same line, just disguised.

Example 3: Just Zero (Normal)

Advanced

Don't get confused! What if ?

Step 1: Add

Step 2: Boomerang

Conclusion:

Getting zero is fine! That's a valid coordinate. The variables did NOT disappear, just equals zero. means the variable itself is gone.

Common Pitfalls

Confusing x=0 with No Solution

If you solve and get , that is a One Solution answer. The solution is on the y-axis. "No Solution" requires the variable to totally vanish (leaving ).

Stopping Early

If you see , finish the math! Move the to get . Be rigorous to check if there is a lurking constant (like ).

Real-Life Applications

Science Experiments:

If you run an experiment twice and get the EXACT same data (Infinite Solutions), your results are reproducible.
If you run it twice and the results contradict each other (No Solution), something is broken or your theory is wrong.