How to Solve a System of Equations on the TI-89

Knowing how to solve a system of equations on the TI-89 saves a page of elimination on every homework set. The TI-89 Titanium gives you three routes to the same answer: solve( with the equations typed as they are, simult( with a coefficient matrix, and rref( on the augmented matrix. Here is how each one works and when to pick which.

Method 1: solve( with and

This is the fastest route and the only one that also handles nonlinear systems. Open solve( from the F2 Algebra menu, type the equations joined with the word and, then a comma and the unknowns as a list in braces.

F21→ solve(

solve(x+y=3 and x-y=1,{x,y})

Type and with the alpha letters, with a space on each side, or pick it from CATALOG. The braces are 2nd ( and 2nd ).

alphaa n d, orCATALOG→ and

2nd(x , y2nd)→ {x,y}

The answer comes back as equations, x = 2 and y = 1 for this system, the same values the matrix methods below give. With three unknowns, add a third equation with another and and list all three variables: {x,y,z}.

For a single equation, solve( takes one variable instead of a list. Command page: solve(.

solve( with one unknown
Input	Result
solve(x^2-2=0,x)	x=−√2 or x=√2
solve(x^2=9,x)	x=−3 or x=3

When there is more than one solution, the results are joined with or. Nonlinear systems, such as a line crossing a circle, come back the same way: one block per intersection point.

Method 2: simult( with a coefficient matrix

For a linear system you can also hand the numbers to simult( directly. The first argument is the coefficient matrix, one row per equation, the second the column of constants. For x + y = 3 and x − y = 1 that is:

simult([1,1;1,-1],[3;1])

simult( is in the Matrix submenu of MATH (2nd 5, then 4) and in CATALOG. The answer is a column: the first entry is x, the second y. Calc89 also accepts linSolve(, which takes the equations as a list and returns the values as a list in the order of the variables.

simult( and linSolve( for the same system
Input	Result
simult([1,1;1,-1],[3;1])	21
linSolve({x+y=3,x-y=1},{x,y})	{2,1}

Both rows solve the system from Method 1 and agree with it: x = 2 and y = 1.

Method 3: rref( on the augmented matrix

Many courses want to see the matrix method written out. Put the coefficients and the constants into one augmented matrix, row reduce it with rref(, and read the solution from the last column. augment( glues the coefficient matrix and the constants together if you have them separately.

2nd54then pick rref(→ rref(

Augmented matrix and row reduction
Input	Result
augment([[1,2][3,4]],[[5][6]])	125346
rref([[1,2,3][4,5,6]])	10−1012

The second row is the system x + 2y = 3 and 4x + 5y = 6. The reduced matrix has the identity on the left, so the last column is the answer: x = −1 and y = 2. If a row turns into zeros with a nonzero last entry, the system has no solution. A row of only zeros means infinitely many solutions with a free variable.

Which method should you use?

Comparison of the three methods
Method	Best for
solve( with and	Fast, nonlinear too, answer as equations
simult( or linSolve(	Linear only, short input from the coefficients
rref( on [A\|b]	Linear only, shows the matrix method, detects no or many solutions

Use solve( when you just need the answer or the system is nonlinear. Use rref( when the assignment asks for the matrix method or when you suspect the system has no unique solution. simult( sits in between: quick for linear systems whose coefficients you already have in a matrix.

The TI-89 Titanium also ships with a Simultaneous Equation Solver app in APPS, which asks for the coefficients in a form. It is a separate handheld app; on the home screen, the three commands above do the same job.

Common mistakes

Stored unknowns. If x or y holds a number, solve( checks numbers instead of solving. Run NewProb (2nd F1) first.
Missing braces. With more than one unknown, the second argument must be a list: {x,y}.
Terms on the wrong side. For simult( and rref(, bring every equation into the form ax + by = c before you copy the coefficients.
A missing coefficient. If a variable is missing from an equation, its coefficient is 0, and the 0 still goes into the matrix.

In Calc89

Calc89 is a CAS calculator app intended for use with TI-89 Titanium. solve( with and, simult( and rref( take the same syntax and return the exact results shown above, and linSolve( is there as an extra. If the system is printed on a worksheet, scan it with the camera and Calc89 types the equations for you. The keystroke tutor then shows the key sequence for the TI-89 Titanium step by step, so you can repeat it on the handheld in class. Calc89 is free to download, Pro features as an in-app purchase.

For the matrix commands in detail, see Matrices on the TI-89: rref, inverse, determinant and eigenvalues.

Frequently asked questions

How do I solve a system of equations on the TI-89 Titanium?

Use solve( with the equations joined by and and the unknowns in braces, for example solve(x+y=3 and x-y=1,{x,y}). For linear systems, simult( and rref( work as well.

Can the TI-89 solve nonlinear systems?

Yes, solve( accepts nonlinear equations joined with and and returns every solution it finds, joined with or. simult( and rref( only work for linear systems.

Why does solve( return false or a strange result?

Usually one of the unknowns holds a stored value. Run NewProb from the Clean Up menu (2nd F1) and try again.

Systems of Equations on the TI-89 Titanium: solve vs. Matrix Method