icon-komplexe-Zahl Math Tutorial: Systems of linear equations

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System of linear equations with two variables x and y

Equating Method and Graphical Solution

a11x+a12y=b1

a21x+a22y=b2

a11= a12= b1=

a21= a22= b2=

Addition Method

a11x+a12y=b1

a21x+a22y=b2

a11= a12= b1=

a21= a22= b2=

Cramer's Rule

In the case of a linear (n × n) equations system with det (A) is equal to 0, the solution can be expressed in the following form:

x=A-1b

xi=1det A a11b1a1n a21b2a2n an1bnann

xi=DiD

The determinant in the numerator D i from D = det A is shown by the i-th column of D is replaced by b.

Cramer's Rule Calculator

a11x+a12y=b1

a21x+a22y=b2

a11= a12= b1=

a21= a22= b2=

Systems of Linear Equations

General representation of a system of linear equations

Generally allows a linear system with m equations and n unknowns by appropriate transformations always bring in the following form:

a11x1+a12x2++a1nxn=b1 a21x1+a22x2++a2nxn=b2 am1x1+am2x2++amnxn=bm

Matrix form of the linear equation system

The linear system of equations can be represented by a coefficient matrix A in matrix notation:

a11a12a1n a21a22a2n am1am2amn x1 x2 xn = b1 b2 bn

or short:

A·x=b

For the definition of the equation system, an indication of the unknown is not required. With the extended coefficient matrix the system of equations can be written as follows:

A|b = a11a12a1n a21a22a2n am1am2amn b1 b2 bn

To find the solution of a linear system of equations, the following three elementary row transformations are useful. The solution set does not change when the following operations:

  • interchanging two rows
  • multiplying a row by a nonzero number
  • adding a line (or multiple of a row) to another row

Solution of the linear system

If the extended coefficient matrix brought to triangular form by means of elementary transformations, the solution can be read directly.

a11a12a1n 0a22a2n 000amn b1 b2 bn

Solution method for linear systems of equations

Addition Method

In the addition method, the equations are added, so that for each addition step, a variable is eliminated. To each one of the equations to be transformed so that the corresponding variable drops out during the addition.

Example: Addition Method

4 · x - 2 · y = 9

4 · x + y = 3

4 · x - 2 · y = 9

8 · x + 2 · y = 6

1 Transformation: Multiply the second equation by 2

12 · x = 15

2 Transformation: addition of equations eliminated the variable y.

x = 15 12 = 5 4

3 Forming: Division by 12 gives the solution the variable x.

y = 4 2 · x - 9 2

4 Transformation: Resolving the first equation for y.

y = 4 2 · 5 4 - 9 2 = 5 2 - 9 2 = - 2

5 Forming: Inserting the solution for x in the equation is the solution for y.

Equating Method

Example: Equating Method

4 · x - 2 · y = 9

4 · x + y = 3

y = 2 x - 9 2

y = 3 - 4 x

1 Transformation: dissolving both equations for y.

3 - 4 x = 2 x - 9 2

2 Forming: Equating equations.

6 x = 3 + 9 2 = 15 2

x = 15 12 = 5 4

3 Forming: Solving for x gives the solution for the variable x.

y = 4 2 · 5 4 - 9 2 = 5 2 - 9 2 = - 2

4. Umformung: Einsetzen der Lösung für x in die nach y aufgelöste Gleichung ergibt die Lösung für y.

Elimination of variables

Example: Elimination of variables

4 · x - 2 · y = 9

4 · x + y = 3

y = 2 x - 9 2

1 Forming: Resolving equation for y.

4 x + 2 x - 9 2 = 3

2 Forming: Insertion in the other equation.

6 x = 3 + 9 2 = 15 2

x = 15 12 = 5 4

3 Forming: Solving for x gives the solution for the variable x.

y = 4 2 · 5 4 - 9 2 = 5 2 - 9 2 = - 2

4 Forming: Inserting the solution for x to y results in the resolution equation, the solution for y.

Gaussian Elimination Method

The Gaussian algorithm is based on equivalent transformations of the system of linear equations. The transformations: row interchange, multiplication of rows with non-zero factors and addition of multiples of a row with a different transform the system of equations to be solved in a simple form.

Calculator for the Gaussian elimination method. The calculator illustrates the method using a 3x3 system of equations. If leading zero coefficients must be prior to use column or rows are exchanged accordingly, so that the leading coefficient by Divison is possible.

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