Mastering 2x2 Equations: A Quick Guide to Cramer’s Rule
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Chapter 1: Introduction to Cramer's Rule
In the realm of linear algebra, the typical method for solving 2x2 equations involves substitution or elimination. However, the technique I will introduce allows you to bypass these conventional steps, making the process significantly more straightforward.
While this method technically requires two steps, you can swiftly arrive at the solution for each variable in just one step per variable. I’ll begin by illustrating this technique with a simple example, followed by an exploration of the underlying mathematics.
The mathematical foundations of this method are quite fascinating, making it worthwhile to delve into them. Let's dive in!
Section 1.1: Solving 2x2 Equations Without Traditional Methods
Consider the following system of equations:
2x + y = 2 x - y = 1
To demonstrate the efficacy of this approach, let's directly find the value of ‘x’:
As promised, we won’t engage in substitutions or eliminations; we will find the answer in one step. If you're uncertain about how I calculated ‘x’, let me clarify the steps taken. We will first focus on the numerator of the fraction I constructed.
Observe the arrows I've indicated. To compute the numerator, we cross-multiply and subtract the coefficients of the non-‘x’ terms, following the order shown by the arrows [(2*(1)) - (1*1)]. For instance, in the equation x - y = 1, the coefficient of x is 1 and the coefficient of y is (-1).
If you're curious why this method works, don’t worry; I’ll provide an explanation later on. Just stay with me for now.
Next, we turn to the denominator. Here, pay attention to the green arrows; we cross-multiply and subtract the coefficients of the ‘x’ and ‘y’ terms, as indicated by the green arrows [((2*(1)) - (1*1))].
At this point, we could substitute the value of ‘x’ back into one of the equations to find ‘y’. However, since I promised to avoid substitution, let’s calculate ‘y’ using the same technique:
The process for calculating the numerator for ‘y’ mirrors the earlier steps; we simply cross-multiply and subtract the coefficients of the non-‘y’ terms, following the same directional arrows. The denominator calculation remains unchanged, leading us to the same result.
Section 1.2: Understanding Cramer's Rule
The technique I just demonstrated is known as Cramer's rule, named after the mathematician Gabriel Cramer from Geneva. The key insight is recognizing that we can represent the 2x2 equation system in a matrix format:
We must assume that this system has a unique solution. One of the main conditions for applying Cramer’s rule is that the determinant [a1b2 - b1a2] must not equal zero. We can then use matrix determinants to compute ‘x’ and ‘y’ as follows:
Notice that the determinant in the denominator for both ‘x’ and ‘y’ is the same. This is why the process to compute the numerator for ‘y’ differs from that for ‘x’.
An observant reader may realize that this method can be extended beyond just 2x2 equations. However, the utility diminishes for systems with 'n' equations and 'n' unknowns, as this would require calculating (n + 1) determinants. For those familiar with determinants of larger matrices, you know they can be quite complex to compute. Therefore, for larger systems, this method is not typically the most efficient.
Chapter 2: Conclusion and Further Insights
Interestingly, Gabriel Cramer was not the first to identify this property in systems of linear equations. A Scottish mathematician named Colin Maclaurin first published on it in 1748, albeit for a special case. Cramer expanded on this concept and applied it to a broader range of unknowns in his work published in 1750.
For solving larger systems of equations, mathematicians often prefer the Gauss elimination method, which I may discuss in a future piece. However, for 2x2 systems, Cramer’s rule remains a valuable addition to your algebra toolkit.
The first video provides a quick overview on how to solve 2x2 systems of linear equations in a straightforward manner, making the concepts easier to grasp.
The second video delves deeper into solving 2x2 systems using substitution and elimination methods, offering a comprehensive guide for learners.
A roadmap of nearly all my work to date. Enjoy!
If you would like to support future content, consider contributing on Patreon. Reference: Gabriel Cramer and Colin Maclaurin. You can read the original essay here.