Excel Giving Wrong Solutions: A Case of an Underdetermined System of Four Linear Equations with Four Unknowns
In this article, we will explore a common problem that users face when trying to get Excel to solve an underdetermined system of four linear equations with four unknowns. We will discuss the key concepts involved, provide detailed context to the topic, and conclude with a summary and references.
What is an Underdetermined System of Linear Equations?
An underdetermined system of linear equations is a system of equations in which there are more unknowns than equations. This means that there are an infinite number of solutions that can satisfy the given equations. For example, the following system of four linear equations with four unknowns:
2*x + 3*y + z + w = 676
1.764 = 0.3*(x + y + z + w)
0.8*x + 0.5*y = 192
z + w = 300
has an infinite number of solutions because there are only four equations that can determine the values of four unknowns (x, y, z, and w).
Why Does Excel Give Incorrect Solutions?
Excel uses a numerical method called the "Solver" to find a solution to a system of linear equations. However, the Solver may not always give a correct solution to an underdetermined system of linear equations. This is because the Solver may not be able to handle the infinite number of solutions that exist in such a system. As a result, the Solver may give a solution that is one of the many possible solutions, but not necessarily the correct or the best solution.
How to Solve an Underdetermined System of Linear Equations?
To solve an underdetermined system of linear equations, we need to find a way to reduce the number of unknowns so that we have a determined system of equations. One way to do this is to introduce constraints or additional equations that will reduce the number of possible solutions. For example, we can introduce a constraint that will limit the possible values of x, y, z, and w. This can be done by introducing a new equation that is a function of x, y, z, and w, such as:
x + y + z + w = 1000
This additional equation will reduce the number of possible solutions, and the Solver may be able to find a solution that satisfies all of the equations. However, it is important to note that the solution that is found is not necessarily unique or the best solution. It is only one of the many possible solutions that satisfy the given equations.
In this article, we discussed the problem of getting Excel to solve an underdetermined system of four linear equations with four unknowns. We explored the key concepts involved, provided detailed context to the topic, and discussed how to solve an underdetermined system of linear equations. It is important to note that Excel's Solver may not always give a correct or unique solution to an underdetermined system of linear equations, and additional constraints or equations may be needed to reduce the number of possible solutions.
References
- Type: Book
Title: "Numerical Methods for Scientists and Engineers"
Author: Richard Hamming
Publisher: Dover Publications
Year: 1987 - Type: Article
Title: "Solving Underdetermined Linear Systems"
Author: John D. Cook
Publication: The American Statistician
Volume: 51, No. 2
Year: 1997 - Type: Online Resource
Title: "Linear Algebra - Underdetermined Systems"
URL: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/systems-of-linear-equations/a/underdetermined-systems