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# HP 32sii Calculator - Solving an Equation

Description
Solve can be used to find the value of any variable in any type of operation.
For example, consider the following equation:
x2 - 3y = 10
If the value of y is known in this equation, then Solve can solve for the unknown x. If the value of x is known, then Solve can solve for the unknown y. This works for "word problems" just as well:
Markup x Cost = Price
If any of these two variables are known, then Solve can calculate the value of the third.
When the equation has only one variable, or when all values are known except one, then to solve for x is to find a root of the equation. A root of an equation occurs where an equality of assignment equation balances exactly, or where an expression equation equals zero. (This is equivalent to the value of the equation being zero.)
Calculator symbol key
The procedures in this document use the following text to represent symbol keys:
 Key Description Text Representation Right shift key RS Left-shift key LS Divide divide
To solve an equation for an unknown variable
1. Press RS, then EQN and display the desired equation.
2. Press RS, then SOLVE, then press the key for the unknown variable. For example, press RS, SOLVE, then X to solve the variable for x. The equation then prompts for a value for every other variable in the equation.
3. For each prompt, enter the desired value:
• If the displayed value is the one desired, press R/S
• If a different value is desired, type or calculate the value and press R/S
Pressing C or R/S can halt a running calculation
When the root is found, it is stored in the unknown variable, and the variable value is viewed in the display. In addition, the X-register contains the root, the Y-register contains the previous estimate, and the Z-register contains the value of the equation at the root (which should be zero).
For some complicated mathematical conditions, a definitive solution cannot be found-- and the calculator displays "NO ROOT FOUND". The following conditions cause the message "NO ROOT FOUND":
• The search terminates near a local minimum or maximum. If the ending value of f (x) (stored in the Z-register) is relatively close to zero, it is possible that a root has been found; the number stored in the unknown variable might be a 12-digit number very close to a theoretical root.
• The search halts because Solve is working on a horizontal asymptote - an area where f (x) is essentially constant for a wide range of X. The ending value of f (x) is the value of the potential asymptote.
• The search is concentrated in a local "flat' region of the function. The ending value of f (x) is the value of the function in this region.
For certain equations, it helps to provide one or two initial guesses for the unknown variable before solving the equation. This can speed up the calculation, direct the answer toward a realistic solution, and find more than one solution, if appropriate.
Example of solving the equation of linear motion
The equation of motion for a free falling object is: d = v0t + 1/2 gt2
Where d is the distance, v0 is the initial velocity, t is the time, and g is the acceleration due to gravity.
Type in the equation:
 Keys Display Description Press LS, CLEAR, ALL, then Y Clears the memory Press RS, then EQN EQN LIST TOP or current equation Selects Equation mode Press RCL, D, RS, [=] RCL, V, X, RCL, T, then [+] D = V x T + _ Starts the equation Press [.], 5, x, RCL, G, x, RCL, T, yx, then 2 T + 0. 5 x G x T^ 2_ Press ENTER D = V x T + 0. 5 x G T Terminates the equation and displays the left end Press RS, then SHOW CK = 6A92 029.0 Performs Checksum and length
G (acceleration due to gravity) is included as a variable so it can be changed for different units (9.8 m/s2 or 32.2 ft/s2).
Example of calculating how many meters an object falls
Calculate how many meters an object falls in 5 seconds, starting form rest. Since equation mode is turned on and the desired equation is already in the display, start solving for D:
 Keys Display Description Press RS, then SOLVE SOLVE _ Prompts for unknown variable Press D V? value Selects D; prompts for V Press 0, then R/S T? value Stores ) in V; prompts for T Press 5, then R/S G? value Stores ) in T; prompts for G Press 9 [.] 8, then R/S D = 122.5000
Example of calculating how long it takes for an object to fall
Using the same equation, how long does it take for an object to fall 500 meters from rest?
 Keys Display Description Press RS, then EQN D = V x T + 0. 5 x G x T Displays the equation Press RS, SOLVE, then T D? 122. 5000 Solves for T; prompts for D Press 5, 0, 0, then R/S V?0.0000 Stores 500 in D, prompts for V Press R/S G?9.8000 Retains ) in V; prompts for G Press R/S SOLVING T=10.1015 Retains 9.8 in G; Solves for T
Example of solving the Ideal Gas Law equation
The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the amount (moles of an ideal gas:
P x V = N x R x T
Where P is pressure (in atmosphere or N/m2), V is volume (in liters), N is the number of moles of gas, R is the universal gas constant (0.0821 liter-atm/mole-K or 8.314 J/mole-K), and T is temperature (Kelvins: K = degrees Celsius + 273.1).
Enter the equation:
 Keys Display Description Press RS, EQN, RCL, P, then X P X _ Selects Equation mode and starts the equation Press RCL, V, RS, [=], RCL, N, X, RCL, R, X, RCL, then T P X V = N X R X T _ Press ENTER P X V = N X R X T Terminates and displays the equation Press RS, then SHOW CK = 13E3 015.0 Performs Checksum and length
Example of calculating an Ideal Gas' pressure
A 2-liter bottle contains 0.005 moles of carbon dioxide gas at 24 degrees Celsius. Assuming that the gas behaves as an ideal gas, calculate its pressure. Since the Equation mode is turned on and the desired equation is already in the display, start by solving for P:
 Keys Display Description Press RS, SOLVE, then P V? value Solves for P; prompts for V Press 2, then R/S N? value Stores 2 in V; prompts for N Press [.], 0, 0, 5, then R/S R? value Stores .005 in N; prompts for R Press [.], 0, 8, 2, 1, then R/S T? value Stores .0821 in R; prompts for T Press 2, 4, ENTER, 2, 7, 3, [.], 1 then [+] T?297.1000 Calculates T (Kelvins) Press R/S SOLVING P = 0.0610 Stores 297.1 in T; solves for P in atmospheres
Example of calculating the density of a gas
A 5-liter flask contains nitrogen gas. The pressure is 0.05 atmospheres when the temperature is 18 degree Celsius. Calculate the density of the gas (N x 28/V, where 28 is the molecular weight of nitrogen).
 Keys Display Description Press RS, then EQN P x V = N x R x T Displays the equation Press RS, SOLVE, then N P?0.0610 Solve for N; prompts for P Press [.], 0, 0, 5, then R/S V?2.0000 Stores .05 in P; prompts for V Press 5, then R/S R?0.0821 Stores 5 in V; prompts for R Press R/S T?297. 1000 Retains previous R; prompts for T Press 1, 8, then ENTER T?18.0000 Calculates T (Kelvins) Press R/S SOLVING N = 0.0169 Stores 291.1 in T; solves for N Press 2, 8, then X 0.4737 Calculates mass in grams, N x 28 Press RCL, V, then divide 0.0947 Calculates density in grams per liter

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