Number of Solutions to a System (DP IB Analysis & Approaches (AA): HL): 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. $0=1$

• 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 $\left[\begin{array}{ccc}1& B_1& C_1\\ 0& 0& 0\\ 0& 0& 1\end{array}|\begin{array}{c}{D}_1\\ D_2\\ D_3\end{array}\right]$

    • assuming $D_2\ne 0$

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

    • $\left[\begin{array}{ccc}1& B_1& C_1\\ 0& 1& C_2\\ 0& 0& 0\end{array}|\begin{array}{c}{D}_1\\ D_2\\ 0\end{array}\right]$

    • where $0=0$ 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 $\lambda$ and $\mu$

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

    • then let $x=\lambda$ and $y=\mu$

  • then find the third variable in terms $\lambda$ and $\mu$ using the equation from the third row

    • e.g. $z=4\lambda -5\mu +6$

  • The general solution is written $x=\lambda$, $y=\mu$ and $z=4\lambda -5\mu +6$

    • where $\lambda \in \mathrm{ℝ}$ and $\mu \in \mathrm{ℝ}$

• In the case where only one row is zero

  • Let the variable corresponding to the zero row be equal to the parameter $\lambda$

    • e.g. if the first row is a zero row then let $x=\lambda$

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

    • e.g. $y=3\lambda -5$ and $z=7-2\lambda$

  • The general solution is written $x=\lambda$, $y=3\lambda -5$ and $z=7-2\lambda$

    • where $\lambda \in \mathrm{ℝ}$