How can I write a system of linear equations?

• To save space we can just write the?coefficients?without the variables

What?is a row reduced system of linear equations?

• A system of linear equations is in row reduced form if it is written as:
  • It is very helpful if the values of?A₁, B₂, C_3?are_?equal to 1

What are row operations?

• • Row operations?are used to make the linear equations?simpler to solve
    • They?do not affect the solution

How can I row reduce a system of linear equations?

How do I solve a system of linear equations once it is in row reduced form?

• Solve the equations?starting at the bottom
  • You have the value for?z?from the third equation
  • Substitute?z?into the second equation and solve for?y
  • Substitute?z?and?y?into the first each and solve for?x

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:

How do I know if the system of linear equations has infinite 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