I used to come across linear algebra problems in math class that seemed impossible to solve. Consider trying to find the value of x, y, and z given this set of two equations:

4x + 3y + z = 8

x + y = 2

If I plug this into Wolfram Alpha, all it can tell me is "y = 4 - x". That doesn't solve for x, y, and z like we want. The problem is intractable.

I used to come across these problems, and I learned a general mathematical principle. In these cases, the only way you can proceed is to **inject new information**. Often this is buried in the initial problem description. So let's say we go back and find a new equation

y + z = 6

Now we plug the three equations into Wolfram Alpha, and voila, we get our solution:

x = -3

y = 7

z = -1

I run across problems like this all the time at work these days. We have an intractable debate, for example:

- We're developing a new technical abstraction; what do we call it?
- We have two competing projects, both with lots of pros and cons. Which should we work on first?
- Some customers like a new experience, and others don't. Should we release it?

You can spend hours in meetings arguing about these types of questions. People can easily get entrenched in supporting the idea they love the most, and the controversy can feel irresolvable. If this goes on longer, people start to identify personally with their position and have their feelings hurt if someone disagrees with them.

But at the beginning, when you first identify the disagreement, there's an opportunity. If a problem seems intractable, **inject new information** and it'll help you solve the problem.

Practically, this means if you're in a meeting and people are repeating the same points, it's time for you to go to find more data. Maybe it's anecdotes, or a Google search of similar problems, or digging into your database to find interesting new data. The time spent finding that information will certainly be more useful than spending time trying to convince people with the same arguments you were using yesterday.

As a final thought, let's go back to the math example from earlier. It had three variables. What if there were four variables to solve for, but just two equations? In that case, I'd actually need to find two more equations. This can apply in real life too. You might need more than one new piece of information to resolve your argument. Discussions are generally iterative, though; adding a new piece of information will move the discussion, even if it doesn't resolve it. It may then make it easier to find the next piece of new information, and you can repeat this process until everyone is in agreement.