Cover image credit: The Grand Challenge Equations: San Diego Supercomputer Center by dullhunk is licensed under CC BY 2.0.

Most of us have experienced the thrill of stepping out into the real world from the academic world as a fresh graduate, full of ambition. Even though you may be saddled with student debt, you believe that your high-paying job will enable you to pay it off in no time, and soon you’d become very rich, and change the world in whatever way you want. You may have some expensive tastes or hobbies, but you’re sure that the income from your new job will allow you to indulge in them without compromising your financial plans.

But can you do a reality check with your financial goals using basic Math?

Let’s put some numbers down for concreteness. Say your take-home pay is 100,000 LCU annually¹. Given your expensive tastes, you estimate that you’re spending 5,000 LCU every month, which translates to an expenditure of 60,000 LCU annually ².

Furthermore, you wish to go from rags-to-riches in about 5 years. Say, you’re currently 200,000 LCU in student debt (and have no real assets to speak of), and your ambition is to grow your net worth from -200,000 LCU currently to, say, 1,000,000 LCU – so you can feel like a million bucks! – using nothing but the excess income left over after your expenses³.

Given all this information, are your plans realistic? Let’s find out.

Let’s start with the fundamental budgeting equation:

$$ \begin{equation} \textrm{Income} = \textrm{Expenses} + \textrm{Savings} \end{equation} $$

Let’s denote annual income by the variable $I$, annual expenses by $E$, and annual savings by $S$. So the budgeting equation now reads as:

$$ \begin{equation} I = E + S \end{equation} $$

In our case, the value of the variable $I$ is 100,000, and that of $E$ is 60,000.

Substituting the values for $I$ and $E$ gives:

$$ \begin{equation} 100000 = 60000 + S \end{equation} $$

Next, let’s quantitatively analyze your ambition to grow your net worth from -200,000 LCU to 1,000,000 LCU in 5 years through your savings. This ambition mathematically translates to:

$$ \begin{equation} S \times 5 = 1000000 - (-20000) = 1200000 \end{equation} $$

We thus have two equations for your annual savings $S$, and all we need to ascertain if your plans are realistic is to check if these equations are consistent, i.e., yield the same value for the same quantity, viz., $S$.

The procedure of determining the value of an unknown variable from an equation is known as “solving the equation”.

In other words, determining what value when substituted for the unknown variable will make the left-hand side (LHS) of the equation equal to its right-hand side (RHS). Solving⁴ eq.(3) for the unknown variable $S$ yields:

$$ \begin{equation} S = 40000 \end{equation} $$

Similarly, solving eq.(4) for $S$ yields:

$$ \begin{equation} S = 240000 \end{equation} $$

So what have we learned from solving for our annual savings $S$ from the two eqs.(3) & (4)?

The budgeting equation implies that based on your income and expenses, your savings will be 40,000 LCU annually, whereas the second equation based on your rags-to-riches ambition implies that your savings should be 240,000 LCU annually. Clearly, these two statements contradict each other, i.e., eqs.(3) & (4) are inconsistent. So you may safely conclude that your financial plans are not realistic.

Thus, you can see how solving equations can help keep you in touch with reality.

In this example, you saw that the equations are inconsistent with each other, i.e., there’s no value of the unknown variable that can satisfy all the equations simultaneously.

There are other situations where you may not be able to solve for a variable, i.e., determine its value from the equations. For example, you know your income from your job is, say 100,000 LCU as before, but since you don’t want to keep a detailed budget to tally your expenses or savings, you don’t know what your expenses and savings are, but would like to compute them using your income information. So your budgeting equation would read:

$$ \begin{equation} 100000 = E + S \end{equation} $$

and you’d like to know the values of $E$ and $S$. Is this possible?

A little thought reveals that it’s not possible as there are several pairs of values for $E$ and $S$ that can satisfy this equation, e.g., $E=50,000$ and $S=50,000$ is one pair of values that satisfies eq.(7), but so does the pair $E=30,00$ and $S=70,000$, as you can easily verify. In fact, there are infinitely many pairs of values for the two variables that can satisfy this single equation. This shows that in order to solve for a given number of variables, we need at least as many equations for them.

In this post, we’ve only scratched the surface of how to solve equations, but there are a myriad of techniques to solve equations of different kinds. Developing facility in solving equations for unknowns is a foundational skill for developing fluency in Math.