Solve:
An introduction to the HP Solve system
Background
The Solve aplet appears in the form outlined here on the HP 38G, HP 39G,
HP 40G, and HP 39g+ graphical calculators but many other HP calculators
offer a similar facility in some form or other. The Solve aplet finds solutions to
equations using successive numerical approximations. Answers are given as
decimal approximations rather than as exact values, and only one variable
can be solved for at a time.
In this text buttons which appear on the keyboard are represented as far as
possible using images of the button:
or
. When the function is ‘shifted’,
appearing above or below a key, then it is written as, for example,
, [T]. Screen menu buttons such as the one in the image
[CLEAR] or
below right are shown as {CAS} or {9CHK}.
Use
The Solve aplet allows the user to store up to ten
equations at a time, with the check mark (9) indicating
which one is currently active. The equation entry screen
is displayed when Solve is activated in the APLET view
by highlighting Solve and pressing {START}. Once the
Solve aplet is active the equation view can be displayed at any time by
pressing
.
When an equation has been entered and marked as
to enter the
active it can be solved by pressing
Numeric view. The variables in the equation are listed in
the order that they appear in the equation. Variable
names must be a single uppercase letter.
An equation is solved by entering known values in all
variables except one, the ‘active variable’, and then
solving for the unknown value by pressing {SOLV}. Only
one value can be solved for at a time.
The Solve aplet begins its search for the solution from
the value in the highlighted field, known as the ‘initial
guess’, when {SOLV} is pressed. Equations with
multiple solutions can be solved by supplying different
initial guesses. The plot view can be used to obtain
these starting points. The left and right sides of the
equation are displayed as two graphs in terms of the
current active variable. If the cursor is moved near to an
intersection point then that value will be transferred as
the initial guess when
is pressed. Which variable is ‘active’ is determined
by the position of the highlight in the Numeric view.
Copyright© 2005, Applications in Mathematics
Some introductory examples
The examples which follow are designed to illustrate the functioning of the
Solve aplet on the HP 39+. Other models operate in a similar fashion1.
Example 1: Exponential growth
The population P, in thousands, of a species of frog at time t years
after the study began is given by the equation P = 5.3e 0.073t + 2.74 .
Find the population after 10 years and the number of years taken
to reach populations of 12,000 and 1500.
•
The equation is entered into the Symbolic view of the Solve aplet.
Press
, highlight Solve using the arrow keys, then press {START}. If
there are any existing equations then they can be deleted using the
key.
Now press
,
,
, [P],
, [T],
, [=],
,
,
,
,
,
,
,
,
, [ex],
,
.
,
,
,
At this point the screen should appear as shown right.
Notice that the most recently entered equation is
always marked as active (9).
•
Change into the Numeric view and enter the known
value of t = 10.
,
,
,
,
.
The highlight is now on P, making it the active
variable. Solve for its value by pressing {SOLV}.
The answer of P = 13.7379 is shown right and
translates to a population of 13,738 frogs.
Copyright© 2005, Applications in Mathematics
•
With the highlight still on P, enter the target population of 12 (ie 12,000).
The highlight will then be on the T field, allowing us to solve for its value.
,
,
, {SOLV}.
This gives an answer of t = 7.64, which translates to
8 years if whole years are required.
•
There is, of course, no solution to the final question. So what happens on
the HP 39 in this situation? Normally we would begin by entering a value of
P = 1.5 and then solving for the value of t.
The calculator will appear to lock up but in fact it is
searching for a valid solution. Because the equation
is asymptotic to the line P = 2.74 as x → −∞ , the
calculator will look further and further along the negative x axis without
success since that is the only way to approach the target of 1.5. Eventually
it will give an answer of –9.999999999E499 which is the largest negative
value the calculator can display or use.
Pressing {SOLV} during the calculator’s search
process will allow you to view the search. The screen
at the right is an example. The negative signs at the
left indicate that it has not managed to ‘bracket’ a
solution. When it is zeroing in on a solution these
change to ‘+’ and ‘-’, although normally this will be too swift to view.
After the process terminates, pressing {INFO} will
result in a screen giving information on what sort of
solution has been found. In this case the result was
“Extremum”, meaning that the calculator was only able
to approach a solution without being able to find one.
“Zero” would imply that it had found a valid solution2.
Copyright© 2005, Applications in Mathematics
Example 2: Probability distribution functions
In this task we will enter the general formula for a cumulative binomial
probability and use it to solve the question below:
A normal pack or cards is shuffled and a card is drawn at random,
recorded and returned to the pack. The process is then repeated to
give a total of 20 draws. What is the probability that you will be
successful in drawing a Heart at least ten but at most 13 times? How
many times must the process be repeated before this probability
rises to at least 20%?
•
Begin by resetting the Solve aplet to remove any current equations or
settings.
Press
•
, move the highlight to “Solve”, press {RESET} and {OK}.
b
n!
n −i
p i (1 − p ) .
i = a ( n − i ) !i !
In the APLET view highlight “Solve”, press {START} to activate the aplet
and then use the sequence of keystrokes below to enter the equation,
using V for the value of the resulting probability.
The equation we will use is: P ( a ≤ X ≤ b ) = ∑
, [V],
, [=],
, [ Σ ],
, [I],
, [=],
, [A],
,
, [B],
,
, [N],
, [ ! ],
,
,
,
, [N],
,
, [I],
,
, [ ! ],
,
, [I],
, [ ! ],
,
,
, [P],
,
, [I],
,
,
,
,
, [P],
,
,
,
, [N],
,
, [I],
,
,
To verify that your equation is correct, press
to highlight the equation
and then press {SHOW}. The result should be as
shown right. Press {OK} to exit the view.
Copyright© 2005, Applications in Mathematics
•
Change to the Numeric view and enter the known values of A=10, B=13,
N=20 and P=0.25 using:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
With the highlight on V, press {SOLV}. After a pause, the result will be as
shown. This probability may seem incorrect but pressing {EDIT} will reveal
the true result of 1.38349051976E-2 or 0.0138. Changing numeric format
in the MODES view to a display using fixed decimal places may give a
better result.
•
To answer the second question, change the value of V to 0.2 and then
solve for N.
,
,
,
,
, {SOLV}
The final result, after longer pause, is shown. It is up to the user to realise
that the required value is actually N=31. The pause is caused by the
iterative solve process which, for this equation, requires a summation to be
done during each iteration.
To verify this value, enter 31 into the N field and solve for the probability V.
,
,
,
, {SOLV}
This results in a probability of 0.2175 which, as required, is “at least 20%”.
1
Other models of HP calculators differ to a greater or lesser degree in their implementation of
Solve. You should consult the manual for your particular calculator for instructions.
2
More information on the {INFO} results can be found in your calculator’s manual.
Copyright© 2005, Applications in Mathematics