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?

• There could be

  • 1 unique solution

  • No solutions

  • An infinite number of solutions

• You can determine the case by looking at the row reduced form

How do I know if the system of linear equations has no solutions?

• Systems with no solutions are called inconsistent

• When trying to solve the system after using the row reduction method you will end up with a mathematical statement which is never true:

  • Such as: 0 = 1

• The row reduced system will contain:

  • At least one row where the entries to the left of the line are zero and the entry on the right of the line is non-zero

    • Such a row is called inconsistent

  • For example:

    • Row 2 is inconsistent if D₂ is non-zero

How do I know if the system of linear equations has an infinite number of solutions?

• Systems with at least one solution are called consistent

  • The solution could be unique or there could be an infinite number of solutions

• When trying to solve the system after using the row reduction method you will end up with a mathematical statement which is always true

  • Such as: 0 = 0

• The row reduced system will contain:

  • At least one row where all the entries are zero

  • No inconsistent rows

  • For example:

    • 

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

• A dependent system of linear equations is one where there are infinite number of solutions

• The general solution will depend on one or two parameters

• In the case where two rows are zero

  • Let the variables corresponding to the zero rows be equal to the parameters λ & μ

    • For example: If the first and second rows are zero rows then let x = λ & y = μ

  • Find the third variable in terms of the two parameters using the equation from the third row

    • For example: z = 4λ – 5μ + 6

• In the case where only one row is zero

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

    • For example: If the first row is a zero row then let x = λ

  • Find the remaining two variables in terms of the parameter using the equations formed by the other two rows

    • For example: y = 3λ – 5 & z = 7 – 2λ