System of Linear Equations

A system of linear equations is a set of algebraic expressions of the type:

a₁₁x₁ + a₁₂x₂ + .....................+ a_1nx_n = b₁

a_21x1 + a₂₂x₂ + .....................+ a_2nx_n = b₂

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

a_m1x₁ + a_m2x₂ + .....................+ a_mnx_n = b_m

• x_i are the unknowns, (i = 1, 2, ..., n).
• a_ij are the coefficients, (i = 1, 2, ..., m), (j = 1, 2, ..., n).
• b_i are the independent terms, (i = 1, 2, ..., m).
• m, n ;        m > n, or, m = n, or, m < n.
• Note that the number of equations need not equal the number of unknowns.
• a_ij and b_i .
• When n takes a low value, it is usual to designate the unknowns with the letters x, y, z, t, ...
• When b_i = 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, regardless of the numbers of equations each system has.

We obtain an equivalent system by eliminating an equation 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.

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

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

4.If an equation in a system is replaced by another equation that results from adding the two 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

Inconsistent

No solution

Consistent

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

It is a system of equations that has 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.