Systems of Linear Equations

MATH 202

SPRING 2002

Solving Systems of Linear Equations
with Graphing Calculators

Suppose we need to determine values for the coefficients A, B, C, and D to satisfy the equation A(x-1)(x²+1) + B(x²+1) + (Cx+D)(x-1)² = x²-4x-4. With a little work, we find that this equation is equivalent to the system

A + C = 0

-A + B - 2C + D = 1

A + C - 2D = -4

-A + B + D = -4

Here are examples showing how to solve this system on the TI-83 Plus, the TI-85, and the TI-89.
On the TI-83, we solve linear systems by converting them to matrix form, and using the rref() command to row-reduce the resulting matrices. The matrix form of the system above is

1 0 1 0 | 0

-1 1 -2 1 | 1

1 0 1 -2 | -4

-1 1 0 1 | -4

We enter this matrix into the calculator using the MATRX menu. To begin, we type [2nd] [MATRX] and use the arrow keys to highlight EDIT. The line for a matrix called [A] is highlighted, so we type [ENTER] to begin editing matrix [A].
The dimensions we want are 4 by 5, so we first enter 4 and 5 along the top line of the screen. Then, using the arrow keys, the number keys, and the [ENTER] key, we fill in the entries of the matrix. Here's what the screen looks like when we have filled in the matrix

When we are finished entering the matrix, we type [2nd] [QUIT] to return to the home screen.
Next, we want the calculator to find the row-reduced echelon form of matrix [A]. The command for doing this is located in the MATRX, MATH menu, so we type [2nd] [MATRX] again, and use the arrow keys to highlight MATH. We use the arrow keys again to scroll down to selection B:rref(, and type [ENTER] to select that.
Back on the home screen, we see rref(. To enter the name of the matrix we return to the MATRX page (type [2nd] [MATRX]) and find that NAMES is already highlighted. In fact, the line 1:[A] is already hightlighted, so we simply type [ENTER] to select it.
The home screen now shows rref([A]. We close the parentheses and type [ENTER] again. The screen shows us the row-reduced echelon form of the matrix [A]. It looks like this:

This tells us that the solution to our system of equations is

A = 2.5, B = -3.5, C = -2.5, D = 2.
If we'd rather not see decimals, we can ask the calculator to convert everything to fractions by typing [MATH] [ENTER] to select ->Frac, and then [ENTER] again.
The result looks like this

and gives us the solution to our system in rational form.
On the TI-85, we use the SIMULT routine to solve a linear system.
To begin, we type [2nd] [SIMULT]. The calculator prompts us for a Number. It wants to know the number of equations we have. Since our system involves four equations and four unknowns, we type 4 [ENTER].
The screen switches to the following display:

The calculator is inviting us to enter the coefficients in the first of the four equations. Using the number keys, arrow keys, and the [ENTER] key, we enter the coefficients as required, so that the screen looks like this:

As soon as we enter the last 0, the screen switches to a prompt for the coefficients in the second equation:

We enter the coefficients for the second equation as before, and then the third and fourth equations.
Here's what the screen looks like after we enter the coefficients for the second equation:

After we enter the coefficients for the fourth equation, we do not get a new prompt, because now the calculator knows all the coefficients. We type [F5] to select SOLVE. The screen tells us that

That is, the solution to our system of equations is

A = 2.5, B = -3.5, C = -2.5, D = 2.
If we want to do further work with these numbers, we need to tell the calculator to save them. We do this by typing [F4] to select STOx (that is, store the x values from the calculation). The calculator prompts us for a name, and we may as well select x (and then [ENTER]).
We type [EXIT] to return to the home screen. At this point, if we type [x-VAR] [ENTER], we get a listing of the results from our calculation above, in vector form. If we want rational numbers, rather than fractions, we type [2nd] [MATH] [F5] [MORE] [F1] (that is, MATH, MISC, MORE, ->Frac ) and then [ENTER]. The screen now looks something like this:

giving us the solution to our original system in rational form.
On the TI-89, there are a few options. We'll use the simult function, which is under the MATH, MATRIX menu.
We type [2nd] [MATH] [4] [5] to get the command simult( to appear in the entry line at the bottom of the screen. We then use the number and punctuation keys as necessary to enter the matrix above. We actually enter two matrices: a four-by-four matrix of coefficients, and a one-by-four matrix of constants. Here is the syntax:

simult([1,0,1,0;^-1,1,^-2,1;1,0,1,^-2;^-1,1,0,1],[0;1;^-4;^-4])

Note that we use commas to separate entries within a matrix row and semicolons to separate the matrix rows themselves.
The calculator solves the system, and displays the system and the solution as follows:

Note that the calculator provides exact answers in rational form. If we want decimal approximations, we press the green diamond key followed by [ENTER], and the calculator provides a decimal approximation of the answer.