Linear Systems

Linear Equation with n Unknowns

It is any expression such as: a1x1 + a2x2 + a3x3 + ... + anxn = b, where ai, b Pertenece ERRE.

Where, ai, are the coefficients, b, the independent term and xi , the unknowns.

Solution of a Linear Equation

Any set of n real numbers that verifies the equation is called a solution to the equation.

Given the equation x + y + z + t = 0, the solutions to it are:

(1, −1, 1, −1), (−2, −2, 0, 4).

Equivalent Equations

Are those that have the same solution.

Systems of Linear Equations

It is a set of algebraic expressions in the form:

a11x1 + a12x2 + .....................+a1nxn = b1

a21x1 + a22x2 + .....................+a2nxn = b2


am1x1 + am2x2 + .....................+amnxn = bm

  • xi are the unknowns, (i = 1, 2, ..., n).
  • aij are the coefficients, (i = 1, 2, ..., m), (j = 1, 2, ..., n).
  • bi are the independent terms, (i = 1, 2, ..., m).
  • m, n Pertenece ENE;        m > n, or, m = n, or, m < n.
  • Note that the number of equations need not equal the number of unknowns.
  • aij and bi Pertenece ENE.
  • When n takes a low value, it is usual to designate the unknowns with the letters x, y, z, t, ...
  • When bi = 0, for all i, the system is called homogeneous.

Solution of a System

It is each set of values that satisfies all equations.

Equivalent Systems of Equations

Equivalent equation systems have the same solution, although they may have a different numbers of equations.

Equivalent systems of equations are obtained by elimination if:

All coefficients are zeros.

Two rows are equal.

A row is proportional to another.

A row is a linear combination of others.

Equivalence Criteria

1 If both members of an equation of a system are added or subtracted by the same expression, the resulting system is equivalent.

2If both members of the equations of a system are multiplied or divided by a number other than zero, the resultant system is equivalent.

3If an equation of a system is added or reduced by another equation of the same system, the resultant system is equivalent.

4If an equation in a system is replaced by another equation that results from adding the equations of a system previously multiplied or divided by nonzero numbers, the resultant system is equivalent.

5 If the order of the equations or the order of the unknowns of a system is changed, it is another equivalent system.

Classifying Systems of Linear Equations

Considering the Number of its Solutions


No solution


It has a solution.

Consistent independent

It has a single solution.

Consistent dependent

The system has infinite solutions.

System of Linear Equations in Triangular Form

They are a system of equations that have an unknown less in each equation than the equation previous.

x + y +   z =  3
    y + 2 z = −1
   ;z = −1

In the 3rd equation, there is z = 1.

Substituting this value into the 2nd equation, it becomes y = 1.

And substituting this into the 1st equation, it becomes x = 3.

x + y + z = 4
   y +  z =  2

With this system, there are more unknowns than there are equations. In this case, take one of the unknowns (eg z) and change its member.

x + y = 4 − z
    y = 2 − z

Consider z = λ , with λ being a parameter to take any real value.

x + y = 4 − λ
    y = 2 − λ

The solutions are:

z = λ   y = 2 λ   x = 2.