IB DP Maths: AA HL复习笔记1.10.2 Algebraic Solutions

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, C₃ 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