Solve system of equations using a graphical calculator

Solve system of equations using a graphical calculator#

A system of equations in the form \(Ax=b\) can be solved with a graphical calculator.

Example

Let’s consider an example

\[\begin{split}\left[\begin{array}{cccccccc}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{4}{1875} & - \frac{1}{625} & -4 & 1\\0 & 0 & 0 & 0 & \frac{1}{625} & \frac{1}{1250} & 1 & 0\\- \frac{4}{1875} & - \frac{1}{625} & -4 & 1 & 0 & 0 & 0 & -1\\\frac{1}{625} & \frac{1}{1250} & 1 & 0 & 0 & 0 & -1 & 0\\4 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\end{array}\right] \left[ \begin{array}{cccccccc} C_1\\C_2\\C_3\\C_4\\C_5\\C_6\\C_7\\C_8 \end{array} \right] = \left[\begin{matrix}0\\0\\- \frac{8}{375}\\\frac{8}{375}\\0\\0\\0\\0\end{matrix}\right]\end{split}\]

  1. Define the augmented matrix \(\left[A|b\right]\) by appending \(b\) on the right of \(A\).

    Example

    The augmented matrix of our example is:

    \[\begin{split}\left[\begin{array}{cccccccc}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{4}{1875} & - \frac{1}{625} & -4 & 1 & -\frac{8}{375}\\0 & 0 & 0 & 0 & \frac{1}{625} & \frac{1}{1250} & 1 & 0 & \frac{8}{375}\\- \frac{4}{1875} & - \frac{1}{625} & -4 & 1 & 0 & 0 & 0 & -1 & 0\\\frac{1}{625} & \frac{1}{1250} & 1 & 0 & 0 & 0 & -1 & 0 & 0\\4 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\end{array}\right] \end{split}\]

    This can be defined in a graphical calculator (TI-84 as an example):

    ../_images/image.png

    Fig. 57 Open the matrix menu: 2nd - matrix#

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    Fig. 58 Go to EDIT#

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    Fig. 59 Edit the first matrix#

  2. Row reduce the matrix, the solution for \(x\) is the right-most column of the matrix.

    Example

    The row reduced matrix looks like this:

    \[\begin{split} \left[\begin{array}{cccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4.375\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -0.006667\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 4.375\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 17.5\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0.0003333\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0.01733\end{array}\right]\end{split}\]

    So:

    \[\begin{split} \left[ \begin{array}{cccccccc} C_1\\C_2\\C_3\\C_4\\C_5\\C_6\\C_7\\C_8 \end{array} \right] = \left[\begin{matrix}4.375\\0\\-0.006667\\0\\4.375\\17.5\\0.0003333\\0.01733\end{matrix}\right] \end{split}\]

    Which can be found on a graphical calculator:

    ../_images/image4.png

    Fig. 60 Use the rref( command in matrix - MATH#

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    Fig. 61 Evaluate rref(A)#

Figures made using https://ti84calc.com/ti84calc