Number of Solutions to a System (DP IB Analysis & Approaches (AA)): Revision Note

Number of solutions to a system

How many solutions can a system of linear equations have?

• A system of linear equations could have

  • 1 unique solution

  • No solutions

  • An infinite number of solutions

• You can determine the case by

  • either looking at the row-reduced form

  • or interpreting the system geometrically

    • e.g. two parallel lines will have no solution

What is an inconsistent system?

• An inconsistent system is one with no solution

• Solving an inconsistent system after using the row reduction method gives

  • a mathematical statement which is never true

    • e.g.

• At least one row will have entries to the left of the vertical line that are zero

  • and entries to the right of the vertical line that are non-zero

    • Such a row is called inconsistent

  • e.g. row 2 is inconsistent in

    • assuming

What is a consistent system?

• A consistent system is one with at least one solution

  • The solution could be unique

  • or there could be an infinite number of solutions

• Solving a consistent system using the row reduction method

  • gives you a mathematical statement which is always true

    • 

    • where is always true

• Note that the row reduced system contains

  • at least one row where all the entries are zero

  • no inconsistent rows

What is a dependent system?

• A dependent system is a consistent system that has an infinite number of solutions

  • Their general solutions can be representing using parameters

How do I find the general solution to a dependent system?

• In the case where two rows are zero

  • let the variables corresponding to the zero rows be equal to the parameters and

    • e.g. if the first and second rows are zero rows

    • then let and

  • then find the third variable in terms and using the equation from the third row

    • e.g.

  • The general solution is written , and

    • where and

• In the case where only one row is zero

  • Let the variable corresponding to the zero row be equal to the parameter

    • e.g. if the first row is a zero row then let

  • Find the remaining two variables in terms of using the equations from the other two rows

    • e.g. and

  • The general solution is written , and

    • where