Systems of equations, Mathematics

Assignment Help:

Since we are going to be working almost exclusively along with systems of equations wherein the number of unknowns equals the number of equations we will confine our review to these types of systems.

All of what we will be doing now can be easily extended to systems along with more unknowns more equations than unknowns if require be.

 Let's begin with the subsequent system of n equations with the n unknowns, x1, x2,..., xn.

a11 x1 + a12 x2 +................+a1n xn = b1

a21 x1 + a22 x2 +.............. +a2n xn  = b2

...................

an1 x1 + an2 x2 +............... +ann xn  = bn                              ...................(1)

Remember that in the subscripts on the coefficients for this system, aij, the i corresponds to the equation which the coefficient is in and the j corresponds to the unknown which is multiplied via the coefficient.

To utilize linear algebra to solve this system we will initially write down the augmented matrix for such system. An augmented matrix is actually just each the coefficients of the system and the numbers for the right side of the system written into matrix form. Now there is the augmented matrix in this system is,

1802_Systems of Equations.png

For solve this system we will utilize elementary row operations that we'll define these in a bit to rewrite the augmented matrix into triangular form. There is the matrix will be in triangular form if all the entries below the major diagonal there is diagonal containing a11, a22, ...,ann, are zeroes.

Once it is done we can recall that all rows in the augmented matrix correspond to an equation. We will after that convert our new augmented matrix goes back to equations and at such point solving the system will turn into very easy.

Before working an illustration let's first describe the elementary row operations. There are three of them.

1.   Interchange two rows. It is exactly what this says. We will interchange row i along with row j. The fact that we'll use to denote such operation is: Ri  ↔ Rj

2.   Multiply row i with a constant, c. it means that all entry in row i will get multiplied with the constant c. The fact for this operation is: cRi

3.   Add a multiply of row i to row j.  Inside our heads we will multiply row i with an suitable constant and after that add the results to row j and place the new row back in row j leaving row i in the matrix unchanged. The fact for this operation is: cRi + Rj

It's all the time a little easier to know these operations if we see them in action.  Therefore, let's solve a couple of systems.


Related Discussions:- Systems of equations

Explain comparing mixed numbers in maths, Explain Comparing Mixed Numbers i...

Explain Comparing Mixed Numbers in maths? A mixed number is made up of two parts: a whole number and a fraction. For example: 2(3/4) 2(3/4) is read "two and three-fourths

Method of reduction of order, Consider the equation x 2 y′′+ xy′- y = 4x...

Consider the equation x 2 y′′+ xy′- y = 4x ln x (a) Verify that x is a solution to the homogeneous equation. (b) Use the method of reduction of order to derive the second

What is common triangles, Q. What is Common Triangles? Ans. Some t...

Q. What is Common Triangles? Ans. Some triangles appear more commonly than others. You will come across two triangles repeatedly as you learn more about trigonometry. T

Linear equation, The sum of the digit number is 7. If the digits are revers...

The sum of the digit number is 7. If the digits are reversed , the number formed is less than the original number. find the number

Objectives of multiplication and division, Objectives After reading t...

Objectives After reading this unit, you should be able to 1. Explain the meaning of multiplication / division and interpret it in different contexts; 2. Convert symbo

Surface areas and volumes, a conical vessel of radius 6cm and height 8cm is...

a conical vessel of radius 6cm and height 8cm is completely filled with water.a sphere is lowered into the water and its size is such that when it touches the size it is immersed.w

#title.square footage, The area of a rectangular yard is 480 square feet. T...

The area of a rectangular yard is 480 square feet. The yard is 24 feet wide. How many feet do I need to fence all four sides?

What is the prime factorization of 84, What is the prime factorization of 8...

What is the prime factorization of 84? This is the only answer choice which has only PRIME numbers. A prime number is a number along with two and only two distinct factors. In

Unitary method, what is the history of unitary method

what is the history of unitary method

Relative frequency definition, Relative Frequency  This type of probab...

Relative Frequency  This type of probability requires us to make some qualifications. We define probability of event A, occurring as the proportion of times A occurs, if we re

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd