Using an array to solve a system of equations

In summary, using an array to solve a system of equations involves representing the equations in a matrix form, typically using augmented matrices. This method allows for the application of various techniques, such as Gaussian elimination or matrix inversion, to find the values of the variables. Arrays simplify calculations and enable systematic manipulation of the equations, making it easier to identify solutions, whether unique, infinite, or nonexistent.
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barryj
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I was given a problem to use an array, not a matrix, to solve a system of equations. I have not heard of doing this and can not find any reference to how to do it.
I was given a problem to use an array, not a matrix, to solve a system of equations. I have not heard of doing this and can not find any reference to how to do it.

like... 2x+3y = 12, and 3x-4y = 14 How is this solved using an array or grid?
 
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Sorry, I can't help. I know a lot about this subject, but I don't know what the difference is between an array and a matrix. They are the same thing to me. Maybe you need to ask for clarification from your instructor?

As an aside, I spent some time in my career as an EE working with other good EEs that were educated in other countries. They knew what they were doing, because you probably wouldn't be working for our company if you didn't. But we were often confused by the jargon each used and their approach to problems. What ALWAYS worked was to go back to the basic physics, then everyone was on the same page.

PS: I think they are referring to Cramer's Rule; OTOH, no, IDK either.
 
  • #3
It may not be obvious but when you solve that problem with a matrix, you are also using vectors. The statement of the problem is ## \begin{bmatrix} 2 & 3\\ 3 & -4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 12 \\ 14 \end{bmatrix}##. That might be what they are meaning.
 
  • #4
barryj said:
I was given a problem to use an array, not a matrix
"Array" is more of a computer science term, where an ordinary array is a list, and a two-dimensional array corresponds to a matrix. AFAIK, "array" not so much a term used in mathematics.
 
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Does anybody ever use determinants to solve a system of linear equations?
 
  • #7
gleem said:
Does anybody ever use determinants to solve a system of linear equations?
Yes. Cramer's rule, which is my first choice for simple systems. Especially if there are zeros in the matrix.

Also especially good if you only need the solution to one of the variables, which is never actually the case, in my experience.
 
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Obviously, if [itex]\mathbf{A}\mathbf{x}=\mathbf{y} [/itex] then [itex]\mathbf{x}=\mathbf{A}^{-1}\mathbf{y} [/itex] as long as [itex]\mathbf{A}^{-1} [/itex] exists. Usually, this just means that [itex]det(\mathbf{A})\neq 0 [/itex].
 

FAQ: Using an array to solve a system of equations

What is an array in the context of solving systems of equations?

An array is a structured arrangement of numbers, symbols, or expressions, typically organized in rows and columns. In the context of solving systems of equations, an array is often used to represent the coefficients of the variables in the equations, allowing for efficient manipulation and computation using methods such as Gaussian elimination or matrix operations.

How can I represent a system of equations as an array?

A system of equations can be represented as an augmented matrix, which combines the coefficients of the variables and the constants from the equations into a single array. For example, the system of equations: 2x + 3y = 5 and 4x + y = 6 can be represented as the augmented matrix: [[2, 3, 5], [4, 1, 6]]. Here, each row corresponds to an equation, and each column corresponds to a variable or the constant term.

What methods can be used to solve a system of equations using arrays?

Common methods for solving a system of equations using arrays include Gaussian elimination, which transforms the augmented matrix into row echelon form, and back substitution to find the variable values. Another method is using matrix operations, such as finding the inverse of the coefficient matrix (if it exists) and multiplying it by the constant matrix to obtain the solution.

What are the advantages of using arrays to solve systems of equations?

Using arrays to solve systems of equations offers several advantages, including the ability to handle large systems efficiently, the use of systematic algorithms for solution, and the facilitation of numerical methods that can provide approximate solutions when exact solutions are difficult to obtain. Arrays also make it easier to implement solutions in programming languages and software for computational mathematics.

What limitations exist when using arrays to solve systems of equations?

One limitation of using arrays is that they require a square matrix (equal number of equations and unknowns) for certain methods like finding the inverse. Additionally, if the system is inconsistent (no solutions) or dependent (infinitely many solutions), the methods may not yield a unique solution. Moreover, numerical instability can occur in computations, especially with large or poorly conditioned matrices, leading to inaccurate results.

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