HP 33s Scientific manual Verifying the Result, 6Solving Equations

When SOLVE evaluates an equation, it does it the same way Xdoes — any "=" in the equation is treated as a " – ". For example, the Ideal Gas Law equation is evaluated as P ⋅ V – (N ⋅ R ⋅ T). This ensures that an equality or assignment equation balances at the root, and that an expression equation equals zero at the root.

Some equations are more difficult to solve than others. In some cases, you need to enter initial guesses in order to find a solution. (See "Choosing Initial Guesses for SOLVE," below.) If SOLVE is unable to find a solution, the calculator displays  ! .

See appendix D for more information about how SOLVE works.

Verifying the Result

After the SOLVE calculation ends, you can verify that the result is indeed a solution of the equation by reviewing the values left in the stack:

„The X–register (press ‡to clear the VIEWed variable) contains the solution (root) for the unknown; that is, the value that makes the evaluation of the equation equal to zero.

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The Y–register (press ) contains the previous estimate for the root. This number should be the same as the value in the X–register. If it is not, then the root returned was only an approximation, and the values in the X– and Y–registers bracket the root. These bracketing numbers should be close together.

The Z– register (press again) contains this value of the equation at the root. For an exact root, this should be zero. If it is not zero, the root given was only an approximation; this number should be close to zero.

If a calculation ends with the  ! , the calculator could not converge on a root. (You can see the value in the X–register — the final estimate of the root — by pressing ‡or bto clear the message.) The values in the X– and Y–registers bracket the interval that was last searched to find the root. The Z–register contains the value of the equation at the final estimate of the root.

„If the X– and Y–register values aren't close together, or the Z–register value isn't close to zero, the estimate from the X–register probably isn't a root.

„If the X– and Y–register values are close together, and the Z–register value is close to zero, the estimate from the X–register may be an approximation to a root.

7–6Solving Equations