Gauss-Jordan Elimination Method

Gauss-Jordan Elimination Method
Gauss-Jordan elimination method is one of the popular approaches of handling complex computation problems. It reduces systems of linear equations into matrices that are easier to compute and solve. Consequently, this method has a wide applicability in the modern world, such as in the field of mathematics, engineering, and computer science. Through the process of discovery learning, this project explores the history of Gauss-Jordan elimination method. A real world activity is also included to highlight how it is applied in the modern world.
Gauss-Jordan Elimination Method Timeline
Video: Using Gauss-Jordan to Solve a System of Three Linear Equations
I selected this video because it covers all aspects of the Gauss-Jordan elimination method for three linear equations. First, the instructor shows the viewer how to turn linear equations into a matrix. Secondly, he shows how the matrix is solved by converting it into a unit matrix. My rationale for selecting this video is that it is clear and precise, allowing for easy comprehension. There is also a step where the instructor switches the first row with the second row to make the calculation easier, which I found very informative. The video can be accessed through this link: /watch?v=CsTOUbeMPUo.
Real World Activity Exploring the Gauss-Jordan Method
Analysis of traffic flow is a potential use of the Gauss-Jordan method. Congestion in the roadways is a significant issue that can lead to a variety of unfavorable results. As a consequence of this, the Gauss-Jordan approach may be used in the process of designing an ideal road network that allows for the smooth flow of traffic. By making use of the information obtained from the flow of traffic, a set of linear equations and an augmented matrix are constructed. Take a look at the following illustration of a typical road network. The streets are only accessible in one direction, as indicated by the arrows that are placed along their edges. The volume of cars that enter and exit the network per hour is used to quantify the flow of traffic in both directions. The flow traffic model would look as follow:
1062188251222400 vphA800 vphDX1X2X3X4C600 vph300 vphB350 vph125 vph225 vph400 vphA800 vphDX1X2X3X4C600 vph300 vphB350 vph125 vph225 vph
A system of linear equations would be generated as follows:
Intersection A: Traffic out will be x1 + x2 = 400 + 225 = 625
Intersection B: Traffic out will be x1 + x4 = 350 + 125 = 475
Intersection C: Traffic out will be x3 + x4 = 600 + 300 = 900
Intersection D: Traffic out will be x2 + x3 = 800 + 250 = 1050
Thus, the linear equations will be as follow:
x1 + x2 = 625
x1 + x4 = 475
x3 + x4 = 900
x2 + x3 = 1050
Calculating a Real World Activity Using the Gauss-Jordan Method
An investor puts $10,000 into three investment accounts. The first account pays 5% interest, the second 8%, and the third 9%. The three investments accrued a total of $770 in the previous fiscal year. How much was invested at each account if the amount invested at 5% was half the amount invested at 9%?
Solution
Let x be the amount invested at 5% interest, y at 8%, and z at 9%.
x + y + z = 10,000
0.05x + 0.08y + 0.09z = 770
2x – z = 0
A=1110....