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

Undamped - forced vibrations, We will firstly notice the undamped case. The...

We will firstly notice the undamped case. The differential equation under this case is, mu'' + ku  = F(t) It is just a non-homogeneous differential equation and we identify h

Solve the subsequent quadratic equation, Solve the subsequent quadratic equ...

Solve the subsequent quadratic equation: Solve the subsequent quadratic equation through taking the square roots of both sides. 3x 2 = 100 - x 2 Solution: Step 1

Probability, There are 20 defective bulbs in a box of 100 bulbs.if 10bulbs ...

There are 20 defective bulbs in a box of 100 bulbs.if 10bulbs are choosen at random then what is the probability of there are just 3defective bulbs

Methods of set representation, I have an assignment of set theory, please E...

I have an assignment of set theory, please Explain Methods of set representation.

Operations and properties, use an expression to write an expression with fi...

use an expression to write an expression with five 3s that has a value of 0

Write down the first few terms of the sequences, Write down the first few t...

Write down the first few terms of each of the subsequent sequences. 1. {n+1 / n 2 } ∞ n=1 2. {(-1)n+1 / 2n} ∞ n=0 3. {bn} ∞ n=1, where bn = nth digit of ? So

Arc Length and Sector Area, how do i find the diameter of a circle if i hav...

how do i find the diameter of a circle if i have the shaded sectors area of 263.76 and the central angle of that circle is 210 degrees?

The volume and surface area of this solid , The region bounded by y=e -x a...

The region bounded by y=e -x and the x-axis among x = 0 and x = 1 is revolved around the x-axis. Determine the volume and surface area of this solid of revolution.

Trigonometry, trigonometric ratios of sum and difference of two angles

trigonometric ratios of sum and difference of two angles

Mensuration, if area of a rectangle is 27 sqmtr and it perimeter is 24 m fi...

if area of a rectangle is 27 sqmtr and it perimeter is 24 m find the length and breath#

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