How To Solve 2×2 Equations In A Single Step

The conventional approach to solve 2×2 equations in linear algebra involves elimination/substitution. However, the method that I will be demonstrating in this essay lets you skip these steps, simplifying the effort in the process.

In a strict sense, this method requires two steps. But you will directly arrive at the result for each unknown, with one step for each unknown. First, I will demonstrate the procedure using a simple example. Later, I will cover the mathematics behind it.

The mathematics behind this method is actually profound. So, it is well worth dipping our toes into it. Without any further ado, let us begin.

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Solve 2×2 Equations Without Elimination/Substitution

Consider the following 2×2 equation system:

2x + y = 2

x − y = 1

To show you the effectiveness of this method, note that we can directly arrive at the value for ‘x’ as follows:

As I had promised, we have no substitutions or eliminations with this procedure; we arrive at the answer in a single step. If you are confused as to how I solved for ‘x’, let me explain what I did there. First, let us concentrate on the numerator of the fraction that I just constructed.

Focus on the black arrows that I’ve drawn. To compute the numerator, what I do is cross-multiply and subtract the coefficients of all the non-’x’ terms in the order indicated by the arrows [(2*(−1)) − (1*1)]. For example, in the equation (x − y = 1), the x-coefficient is 1 and the y-coefficient is (−1).

If you are wondering why we have to do it this way, don’t worry; I’ll explain it later in the essay. Just bear with me for now.

Next, let us move on to the denominator. This time, focus on the green arrows; I cross multiply and subtract the coefficients of the ‘x’ and ‘y’ terms in the order indicated by the green arrows [((2*(−1)) − (1*1)].

At this stage, we could just substitute the ‘x’ value in one of the equations to get the ‘y’ value. But since I promised that we would avoid substitution, let me solve for ‘y’ using the same method:

The execution for the numerator in the y-value computation is analogous to what I did before; I just cross-multiply and subtract the coefficients of all the non-y terms in the order indicated by the black arrows. The procedure for the denominator remains the same as before and we get the same result as well.

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How to Solve 2×2 Equations in a Single Step — The Rule

The method that I just illustrated is known as Cramer’s rule, named after Genevan mathematician Gabriel Cramer. The trick is to realise that we are dealing with rows and columns in a 2×2 equation system.

Implicitly, we first transform the equation system into the following matrix form:

Next, let us assume that the system of equations has a unique solution. In other words, one of the primary conditions for applying Cramer’s rule is the result for [a1b2 − b1a2] is non-zero. Finally, using the concept of matrix determinants, we can compute ‘x’ and ‘y’ as follows:

As you can see, the determinant on the denominator for both ‘x’ and ‘y’ are the same. This is also why the procedure to compute the numerator for ‘y’ varies from that for ‘x’.

The prudent reader might have noticed that this method can be extended beyond just a 2×2 system of equations. However, it probably stops being useful because for ’n’ equations with ’n’ unknowns, this method requires us to compute (n + 1) determinants.

If you ever computed determinants of higher-order matrices, you’d know that they can be computationally intensive. So, for larger systems of equations, this method is not usually the most efficient.

Final Thoughts

Gabriel Cramer was actually not the first person to note this property of systems of linear equations. A Scottish mathematician named Colin Maclaurin published about it first (albeit a special case) in 1748. Cramer generalised the notion and extended the rule to an arbitrary number of unknowns in his work published in 1750.

To solve larger systems of equations, mathematicians often prefer a method known as the Gauss elimination method, which I might cover in a future essay. For 2×2 equation systems though, Cramer’s rule is a useful tool to have in your algebra toolkit.

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Reference: Gabriel Cramer and Colin Maclaurin.

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Further reading that might interest you:

• How To Really Treat Percentages With Negative Numbers?
• The Strong Law of Small Numbers.
• 3 Reasons Why Deep Space Travel Is Really Challenging

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