Short Communication - Journal of Applied Mathematics and Statistical Applications (2018) Volume 1, Issue 2

# A new approach for solving linear equations with first order through derivatives

- *Corresponding Author:
- Rami Obeid

Head of data management and analysis division Central bank of Jordan, Jordan

**Tel:**962-795-855-036

**E-mail:**Rami.obeid3@gmail.com

**Accepted date:** September 17, 2018

**Citation: **Obeid R. A new approach for solving linear equations with first order through derivatives. J Appl Math Statist Appl. 2018;2(1):8-10.

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## Abstract

This paper proposes a simple method to solve the first order linear equations, the proposed method is equivalent to classical Cramer’s rule for solving general systems of 2 linear equations, then it describes if there is a relationship between this method and the derivatives. The results show that there is a possible relationship between the method presented in this paper and the derivatives. Furthermore, we can use the first derivative to solve linear equations with first order.

### Keywords

Linear equations, Matrix, First derivatives, Cramer’s rule.

### Introduction

There are various methods to solve the linear equation, The Cramer's rule is the most common of these methods [1], Klein [2]
described the approach based upon Cramer’s rule, the of the linear equation system can be written in matrix form: A*x =b* , Cramer’s
rule is efficient in solving systems of 2 linear equations. Some recent developments of using Cramer’s rule described in some papers,
these papers can be found in [3-5] and the references therein.

**Solving linear equations**

First, this paper introduces a simple method for solving general systems of 2 linear equations, and we will prove it using Cramer's rule as following:

**Proof: **We can prove the above rule by using Cramer's rule, as we know when we use Cramer's rule we find that:-

(1)

and (2)

First, we want to prove that

Similarly, we can prove also that by using the above method.

**The possible relation between rule 1 and the first derivative: -**

Now we will discuss if there is a relation between rule 1 and derivatives: -

Using the same equation in rule (1):-

We can represent and by the following matrix:-

Thus, if we want to find the values of x and y we can easily reach to the same results in Rule 1, where the two columns that we used to find x in rule 1 is similar to the coefficients of the constants and the variable y in the matrix of respectively:-

Similarly, the two columns that we used to find y in rule 1 is similar to the coefficients of the variable x and the constants in the matrix of respectively: -

**Solving linear equations with first order by first derivatives**

To explain how to solve linear equations with first order by first derivatives; suppose we have the following linear equation system: -

x + 2y = 4

3x − y =5

Let

49x =98⇒x = 2

Now to find the value of Y:-

49y = 49⇒y =1

### Conclusion

We have studied a simple method for solving systems of 2 linear equations. The method can be easily applied to systems of 2 linear equations. Also, we have described if there is a relationship between this method and the first derivative, the paper show that there is a possible relationship between them, and we can solve linear equations with first order by first derivatives.

## References

- Cramer G. Introduction l?Analyse des lignes Courbes algbriques. Geneva: Europeana. 1750; pp 656-9.
- Klein RE. Teaching linear systems theory using Cramer?s rule. IEEE Transactions on Education, 1990; 33:258-67.
- Diaz-Toca GM, Vega GL, Lombardi H. Generalizing Cramer?s Rule: Solving Uniformly Linear Systems of Equations. SIAM J. Matrix Anal. & Appl. 2005; 27:621-37.
- Habgood K, Arel I. A condensation-based application of Cramer?s rule for solving large-scale linear systems. J Discrete Algorithms. 2012; 10: 98-109.
- Kyrchei II. Cramer?s rule for quaternionic systems of linear equations. Journal of Mathematical Sciences. 2008; 155:839-58.