Simultaneous solution of multivariable equations: a leader’s dilemma

It’s rare that I turn to mathematics as an inspiration of leadership lessons in my essays, but today I have been reminded of the challenge of solving complex problems with multiple variables at play… and needing to solve for those variables at the same time… or simultaneously.  I encountered this topic back in college with my first go at Linear Algebra… and worked on it more deeply in grad school.  In those days, before pc’s and broad-based personal computing, we used the “matrix method” and hand-built matrices to solve theoretical equations manually.  A few definitions follow below will help build a base of understanding so we can turn to how this concept is a basis for a leadership lesson.

 

Simultaneous solutions for multivariable equations require finding the values for multiple unknown variables that satisfy all equations at the same time, using substitution, elimination, or matrices

Core Methods

• Substitution Method:
• Isolate one variable in one equation.
• Plug that expression into the other equations.
• Solve for the remaining single variable.  
• Elimination Method:
• Multiply equations by numbers so one variable has matching or opposite coefficients.
• Add or subtract the equations to cancel out that variable.
• Repeat to peel back layers until one variable is left.  
• Matrix / Linear Algebra:
• Convert the system into row-echelon form or use inverse matrices.
• Best for large systems with three or more variables.  

 

 

Now that we have some of the mathematical definitions in hand, how in the world do they relate to leadership and lessons on decision making and problem solving for leaders?  Over the past few weeks, in conversations with a number of C-suite execs, I keep running into individuals facing challenging moments and tough decisions looking for my advice. These execs span a variety of companies and industries, but all are facing very difficult issues with many variables affecting the moments at hand.  As I dig into the conversations with each individual, we start going through the wide range of factors (think “variables”) that need to be sorted out in order to make the best decision possible.  It was after one of these discussions and leaving the “zoom call,” it dawned on me that here was another example of an executive trying to solve a “multivariable equation” and that they and I were informally using one of the methods ( mentioned above) without realizing it and without much positive impact.  We “swirled” around the problem/issue/challenge and debated the various factors/decisions/”variables” that would need to be considered…. but we made limited progress.

As a followup for one of those calls, I took some time and wrote out the factors/ ”variables” at hand (there were 7) and did my own ranking of the 7 factors/”variables” on a scale of “importance.”  Now “importance” is extremely subjective, so I looked to add a little analytic discipline of my own and de-composed it into three elements… “Importance” = ”Strategic impact on the business” X “Impact on me and my family” X “Impact on the organization.”  This allowed me to thus “score” each of the factors/ ”variables” and build a prioritized list of the 7.  In the same fashion, I asked the exec to do the same… NOT try to problem solve all 7 factors/ ”variables” at once but to use this technique to “prioritize” the list.  While we had a little debate on the order of #3-#7, the top two, most “important “ factors/”variables” were clearly at the top of both of our lists.

Once we had the top 2 identified, I encouraged that we make the decisions on these two factors/ “variables” first… discuss/debate then make the call on the top 2 …then and only then decide on the other 5 factors/”variables” once we had the top 2 set in stone… in other words we used a “prioritized method” to get all 7 factors/”variables” sorted and thus were able to chart the path forward guided by making the decisions on the most important “variables” first!

This technique was helpful to navigate the complicated, “multi-variate” problems facing these execs, and it reminded me that finding new approaches to problem solving and decision making is important and can come from lessons learned in a college/grad-school calculus class!