Linear Equations

Consistent or Inconsistent

consistent and independent

any number works consistent and dependent

ConsistentInconsistentDependentIndependent
There is a solutionThere is no solutionInfinite solutionsUnique Solution
Test Question

A linear equation has 3 options: no solution, unique solution, infinite number (no finite number over 1 and under infinity)

System of Linear Equations

Example 1

If any two rows are linearly dependent the whole system has infinite solutions. In this example only the and rows are dependent. Since a solution exists it is also consistent.

  • Def. Linear combinations: multiply add or divide equations by constants to combine two equations into one.

Dependent and consistent

Test Question

Suppose a SLE is dependent. Does that SLE have infinite solutions? (yes) Is anything a solution? (no)

Example 2

Once it is inconsistent then the rest doesn't really matter, but in this case since none of the rows cancelled out this is independent.

Inconsistent and independent

Inconsistent, Consistent, Dependent, Independent

I/CI/DCriteria
InconsistentIndependentAt least one row is inconsistent and none are dependent
InconsistentDependentAt least one row is inconsistent and at least one is dependent
ConsistentIndependentAt least one row is consistent and none are dependent
ConsistentDependentAt least one row is consistent and at least one is dependent