TI-83: Complex 3 equations in 2 variables, 2 parallel, no unique solution

Complex 3 equations in 2 variables, 2 parallel, no unique solution

The main page for solving systems of linear equations on the TI-83 and TI-83 Plus.
The previous example page covers a Complex 3 equations in 2 variables, non-intersecting.
The next example page covers a Complex 3 equation 2 variable situation, where there is a solution.

WARNING: The TI-83 and TI-83 Plus are almost identical in terms of the material presented here. The major difference is the labels that are on certain keys. The TI-83 has a

key, whereas on the TI-83 Plus requires 2 keys to achieve the same result, namely, the

key. The text below will be done from the perspective of the TI-83. That is, all reference to the MATRIX key will be demonstrated via the

key. If the user has a TI-83 Plus then the key strokes should be

. To save some space, and to ignore this difference, the numeric keys (the gray ones) have been changed in some places to show the key face, as in

. In addition, the

key may be shown as

and the

key may be shown as

, again to save space.

The problem we will use on this page is 5x - 3y = 26
-2x + y = -10
5x - 3y = 17 As was the case in the previous example, we have just two variables, but we have 3 equations. Our use of the rref function has been in situations where the number of equations is equal to the number of variables. In a real sense, we have an extra equation here. If we only had two equations then we could use rref to find their point of intersection. Let us move ahead in that direction. At first we will restrict our analysis to the first two equations: 5x - 3y = 26
-2x + y = -10

Figure 1	Move to the EDIT tab of the MATRIX menu. Again, we will use matrix [A] for this problem.
Figure 2	Matrix [A] has the correct dimensions for solving a system of two equations in two variables, namely 2 rows and 3 columns. For Figure 2 we have gone forward and entered the coefficients and constants from the first two equations.
Figure 3	In Figure 3 we have quit the matrix editor, recalled the command that we want to use, and performed the command. The resulting matrix yields the solution x=4 and y=^–2.
Figure 4	Having found a point of intersection for the first two equations, we can return to the matrix editor and change the first row to hold the values from the third equation. The changed matrix, shown in Figure 4, represents the third and second equations.
Figure 5	In Figure 5 we have quit the matrix editor, recalled the command that we want to use, and performed the command. The resulting matrix yields the solution x=13 and y=16. These values, being different from the coordinates of the solution to the first two equations, are enough to determine that there is no unique solution to the three equations. However, as in the previous example, it might be interesting to see the solution to the first and third equations.
Figure 6	To do this we return to the matrix editor where we replace the second row with the coefficients and constant from the first equation.
Figure 7	In Figure 7 we have quit the matrix editor, recalled the command that we want to use, and performed the command. The resulting matrix indicates that there is no solution to this pair of equalities. The second row now represents the impossible equation 0x + 0y = 1 Returning to the original equations, and focusing on the first and third, we note that these two will graph as parallel lines. The second equation intersects the other two, but at (4,^–2) for the first and (13,16) for the third.
Figure 8	In Figure 8 we have the Y= screen where we have entered all three equations.
Figure 9	An initial graph of the three equations, in the ZOOM Standard settings, shows the two parallel lines and a line cutting across them (a transverse line).
Figure 10	Knowing the two points of intersection we can adjust, as shown in Figure&10, the WINDOW settings to show both points.
Figure 11	This is merely the new graph, in the new WINDOW settings, showing the two parallel lines (equations 1 and 3) and the transverse line (equation 2).