Linear Systems
Linear Equation with n Unknowns
It is any expression such as: a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + ... + a_{n}x_{n }= b, where a_{i}, b .
Where, a_{i}, are the coefficients, b, the independent term and x_{i} , 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:
a_{11}x_{1} + a_{12}x_{2} + .....................+a_{1n}x_{n} = b_{1}
a_{21x}_{1} + a_{22}x_{2} + .....................+a_{2n}x_{n} = b_{2}
...............................................................
a_{m1}x_{1} + a_{m2}x_{2} + .....................+a_{mn}x_{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, 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
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
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.