Systems of Linear Equations

Calculator

System of linear equations with two variables x and y

Equating Method and Graphical Solution

${a}_{11}x+{a}_{12}y={b}_{1}$

${a}_{21}x+{a}_{22}y={b}_{2}$

a11= a12= b1=

a21= a22= b2=

${a}_{11}x+{a}_{12}y={b}_{1}$

${a}_{21}x+{a}_{22}y={b}_{2}$

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:

$x={A}^{-1}b$

${x}_{i}=\frac{1}{\mathrm{det A}}\left|\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{1}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{n1}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{n}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{nn}\end{array}\right|$

${x}_{i}=\frac{{D}_{i}}{D}$

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

${a}_{11}x+{a}_{12}y={b}_{1}$

${a}_{21}x+{a}_{22}y={b}_{2}$

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:

$\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\dots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{2n}{x}_{n}={b}_{2}\\ ⋮\\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\dots +{a}_{mn}{x}_{n}={b}_{m}\end{array}$

Matrix form of the linear equation system

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

$\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{m1}\phantom{\rule{1em}{0ex}}{a}_{m2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)=\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{n}\end{array}\right)$

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:

$\left(A|b\right)=\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{m1}\phantom{\rule{1em}{0ex}}{a}_{m2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}\right|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{n}\end{array})$

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.

$\left(\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}{a}_{12}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ 0\phantom{\rule{2em}{0ex}}{a}_{22}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ 0\phantom{\rule{1em}{0ex}}0\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}0\phantom{\rule{1em}{0ex}}{a}_{mn}\end{array}\right|\begin{array}{c}{b}_{1}\\ {b}_{2}\\ ⋮\\ {b}_{n}\end{array})$

Solution methods for linear systems of equations

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.

$4·x-2·y=9$

$4·x+y=3$

$4·x-2·y=9$

$8·x+2·y=6$

1 Transformation: Multiply the second equation by 2

$12·x=15$

2 Transformation: addition of equations eliminated the variable y.

$x=\frac{15}{12}=\frac{5}{4}$

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

$y=\frac{4}{2}·x-\frac{9}{2}$

4 Transformation: Resolving the first equation for y.

$y=\frac{4}{2}·\frac{5}{4}-\frac{9}{2}=\frac{5}{2}-\frac{9}{2}=-2$

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

Equating Method

Example: Equating Method

$4·x-2·y=9$

$4·x+y=3$

$y=2x-\frac{9}{2}$

$y=3-4x$

1 Transformation: dissolving both equations for y.

$3-4x=2x-\frac{9}{2}$

2 Forming: Equating equations.

$6x=3+\frac{9}{2}=\frac{15}{2}$

$x=\frac{15}{12}=\frac{5}{4}$

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

$y=\frac{4}{2}·\frac{5}{4}-\frac{9}{2}=\frac{5}{2}-\frac{9}{2}=-2$

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

$4·x-2·y=9$

$4·x+y=3$

$y=2x-\frac{9}{2}$

1 Forming: Resolving equation for y.

$4x+2x-\frac{9}{2}=3$

2 Forming: Insertion in the other equation.

$6x=3+\frac{9}{2}=\frac{15}{2}$

$x=\frac{15}{12}=\frac{5}{4}$

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

$y=\frac{4}{2}·\frac{5}{4}-\frac{9}{2}=\frac{5}{2}-\frac{9}{2}=-2$

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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