What's most important method of solving systems of equations? substitution, elimination, or matrix? Why?

Question

What's most important method of solving systems of equations? substitution, elimination, or matrix method? Why?

Which is method is used most often in professions that work with numbers and calculations a lot?

This question leads me to one more question - are systems of equations even used in real world science and technology professions? which fields for example?

Answer 1

Engineers use matrix based methods, but as another answer points out these are all basically the same (Gaussian elimination is a matrix method but also as the name implies an elimination method). In practice the method people use depends on the structure of the system of equations / matrix. See for example the documentation for Matlab's matrix division operator: http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/mldivide.html

Matlab uses an LU decomposition with pivoting, which is something like Gaussian elimination but more numerically stable.

As for who needs to solve systems of linear equations, really everyone in science and engineering. To give a basic example, fitting lines and curves to sets of points usually invovles solving a system of linear equations.

Answer 2

All methods have their merits: substitution is an all-purpose method, because it works for non-linear equations as well; elimination is the fastest, but is not applicable to that large a set of systems; and the matrix method is easy to generalize, and is the same no matter how many variables. Professionals use the matrix method most, since that is the easiest as far as computational complexity goes. Systems of equations are constantly used in the real world; Einstein's theory of relativity is one example of a (very complex) system of equations.

Steve

Answer 3

All those methods are essentially the same and require the same amount of work. For example, what do you have to do to solve a system of equations using matrices? You have to find the inverse of the matrix. But the way you find the inverse is by using Gaussian elimination. (Except maybe for 2x2 matrices, where you can memorize the formula for the inverse.)

Answer 4

Multiply it with the help of '-2'. 3x - y = 3 ---> Multiply with the help of '-2' -2x + 2y = 6 -6x + 2y = -6 -2x + 2y = 6 ------------------ Subtract -4x = -12 x = 3. substitute this 'x' value in the two of the unique equations to get the 'y' value. Edit : you does no longer multiply with the help of three. You 'substitute'. 3x - y = 3 ---> substitute the value of 'x'. 3(3) - y = 3 9 - 3 = y ---> y = 6 So answer is 6.(0.33 determination)

Answer 5

The best method should b substitution. bcause it is much accurate.

May b substitution & elimination. These are used vastly

They r used in programming,theorems everywhere