How to Solve Equations on a TI-84

A TI-84 solves equations in two quite different ways, and choosing the wrong one wastes the most time. The numeric Solver, reached from the MATH menu, handles a single equation with a single unknown and answers it with a number. The PlySmlt2 app, reached from the APPS menu on the TI-84 Plus CE, handles systems of linear equations and answers them all at once.

Neither tool works on an equation exactly as it is written in the question. That is the part students miss. The Solver needs the equation rearranged so that the two sides are equal to a single expression, and the systems app needs the coefficients, not the equations themselves.

This page takes you through both, starting with a one-variable equation you could also solve by hand so that the calculator’s answer is easy to verify.

The Solver: one equation, one unknown

Press MATH and choose the Solver. On the TI-84 Plus it is the last entry in the menu, 0:Solver. On a TI-84 Plus CE running a recent operating system the same tool sits at the bottom of the MATH menu and is labelled Numeric Solver.

The Solver screen opens on a line that begins eqn:0=, and this is where the rearrangement matters. The calculator is looking for a value that makes the expression on the right-hand side equal to zero, so you have to move everything to one side first.

Take 2X − 3 = 5. Subtract five from both sides and the equation becomes 2X − 8 = 0, or in the form the calculator wants, 0 = 2X − 3 − 5. Type 2X-3-5 on the eqn: line and press ENTER.

Rather than solving the rearranged equation directly, it helps to see what the calculator has actually done: it is finding the value where Y1 = 2X − 3 − 5 crosses the axis.

That crossing is exactly what 2:zero finds, which is why the two methods agree. The Solver is simply a faster route when the rearranged expression is awkward to type into the Y= editor.

What the guess prompt is for

After the equation is entered, the cursor blinks beside X= and asks for a value. This is the guess, and it is not a formality. The Solver searches numerically, starting from the number you give it and walking towards a solution, so the guess decides where the search begins.

A guess far from any root, or one that sends the search down a slope that never reaches the axis, wastes time or fails. A guess based on a rough sketch of the graph, or on a couple of rows of the TABLE screen, is reliable. Press ALPHA ENTER to run the search once the guess is in place.

The left-rt figure on the result line is the left side of the equation minus the right side, evaluated at the solution. Zero means the two sides balance exactly there. A small non-zero figure means the search stopped very close to a root rather than precisely on one.

One solution at a time

The Solver returns a single value, so an equation with two roots needs two runs. For X² − 5X + 6 = 0 the roots are 2 and 3. Enter the expression, guess a value near zero and solve, and the calculator reports 2. Change the guess to something above the midpoint and solve again, and it reports 3.

This is also why a suspicious answer is worth a second look. A single root from a quadratic, with no attempt at the second, is an easy mark to drop.

Linear systems with the PlySmlt2 app

For two or more equations in two or more unknowns, the Solver cannot help, because it expects one variable. The route is the PlySmlt2 app, which is preloaded on the TI-84 Plus CE and downloadable from Texas Instruments for older models.

1. Press APPS and choose PlySmlt2 from the list.
2. Press any key to clear the information screen and reach the main menu.
3. Select Simultaneous Eqn Solver.
4. Choose the number of equations and unknowns, then press NEXT.
5. Enter each coefficient and constant, pressing ENTER after every entry to move along the row.
6. Run SOLVE and read the values of the unknowns.

Take the pair Y = 2X + 1 and Y = −X + 4. Typed into the app as two equations in two unknowns, the coefficients go in as 2 and 1 for the first, and −1 and 4 for the second.

The values the app returns are the coordinates of that crossing, so a graph is a quick way to check that you entered the coefficients in the right order. A sign lost on the way in is the most common way a correct method produces a wrong answer here.

What the exam rules say about these tools

The Solver and PlySmlt2 are numeric features, not a computer algebra system, so they sit on the permitted side of the rules for most papers. The restrictions are about the class of calculator rather than the individual menu items.

• The SAT permits a graphing calculator on all of the maths questions and prohibits calculators with a computer algebra system.
• The AP Calculus exams prohibit a CAS, while the TI-84 Plus family is a permitted graphing calculator for those papers.
• The IB requires a graphic display calculator for the papers that allow one, and CAS capabilities are restricted on some papers, so the details depend on the subject and the arrangements your school has with the organisation.

Our AP calculator policy and IB calculator policy pages set out the conditions. Rules are revised from time to time, so confirm the current wording with the board and your teacher before you rely on any particular function.

If a value the Solver reaches is outside the domain of an expression in the equation, the usual message is ERR: DOMAIN, and a malformed entry on the eqn: line produces ERR: SYNTAX, where 2:Goto points straight at the offending character. Once you have a solution, plotting the curve or reading the intercept gives you a second opinion on the answer.