Real Answer has a special window-- the Simultaneous Equation Set window-- that allows us to find a set of values of variables that will satisfy several equations simultaneously. Choose “Simultaneous Equation Set” from the “Windows” menu. First we need to remove anything that had been left in that window. Anything left in the Simultaneous Equation Set window effects its answer, unlike the Answers and Scratch window which responds to only a single line at a time. With the Simultaneous Equation Set window in front, choose “Select All” from the “Edit” menu (or type “a” while holding down the “⌘” button). Then press the “delete” key. Enter the following equations as the sole text in the Simultaneous Equation Set window:
y=x*x {a parabola}
y=2*x+3 {a line}
(You can copy the lines right out of this help window and paste them into the Simultaneous Equation Set window.) Choose “Solve Eq. Set” from the “Solve” menu (or type “r” while holding down the “⌘” button). An answer is presented in the Answers & Scratch window. The x and y values correspond to a point where the line intersects the parabola. Notice that as it solves the problem, Real Answer displays a search window showing the progress towards a solution as it iteratively closes in on an answer. In one line, Real Answer will show the values it plugs into some of the variables as an estimate of the answer set. Then Real Answer will try another set of values, based on how well the first set fit your equations. On the next line in the Solution Search window, Real Answer will display that next set of values tried. This goes on until an accurate answer is found or Real Answer gives up. By choosing the “Iterations” item in the “Solve” menu, you can control criteria that determine when Real Answer will stop trying to find a solution.
There are actually two points which satisfy both equations that we entered:
x y
---- ----
-1 1
3 9
If you first got the (-1, 1) result, find the other by typing “x=4” into the Answers & Scratch window. Hit “enter”. x is set to 4 now. Now hit “⌘R” (or choose “Solve Eq. Set” from the “Solve” menu) and Real Answer will find the (3, 9) pair. If you first got the (3, 9) result, you can set x=-2 in the Answers & Scratch window, then hit “⌘R” to find the (-1, 1) result. There are two important things we’ve seen here:
• Sometimes there is more than one answer to a problem, even though Real Answer only finds one at first.
• Manipulating the initial values of variables can affect which answer is found.
In fact, changing the initial values can frequently allow Real Answer to find an answer when it couldn’t find one at all before.
You can also initialize a variable in the Simultaneous Equation Set window. That has the advantage that every time you Run the solver (e.g. type “⌘R”) the variables always start from the same value, regardless of any previous setting of the variables. Let’s try that. Add a third equation to the Simultaneous Equation Set window, so it looks like:
y=x*x {a parabola}
y=2*x+3 {a line}
y=-12
Now type “⌘R”.
Oops! We get an error message complaining that there are more variables than equations to solve. Real Answer needs some way of distinguishing between equations that need solving and variable initializations present in the Simultaneous Equation Set window. There is a way; we will use a different kind of equals sign to perform variable initialization in the Simultaneous Equation Set window. Add a “:” to the last equation in the Simultaneous Equation Set window so the last equation looks like:
y:=-12
Now choose “Solve Eq. Set” from the “Solve” menu (or type “⌘R”) again.
Real Answer is happy and finds the closest solution. You can use the “:=” equals sign in the Answers and Scratch window, too. If you are familiar with PASCAL, you will recognize that we have borrowed the meaning of the “:=” assignment operator; in Real Answer, also, it just copies the value from the right into the left. You can use expressions on the right, too, even including previously initialized variables. It also doesn’t matter in the Simultaneous Equation Set window if you put an initialization before or after the “=” equations you are trying to solve; all initializations are executed before the iterative solution search begins. For instance this set of equations:
y=x*x {a parabola}
y:=-sqrt(144)
y=2*x+3 {a line}
is exactly equivalent to the previous set.
We have also borrowed the idea of enclosing comments with braces from PASCAL.
The following set of equations should be essentially equivalent to the previous sets:
x=sqrt(y) {the same parabola, or at least half of it}
y:=-sqrt(144)
y=2*x+3 {a line}
Try solving it by replacing everything in the Simultaneous Equation Set window with the above 3 lines and “Solve Eq. Set” from the “Solve” menu.
You get an error message because an intermediate step in the solution search tried using a value of -12 for y, and the square root of a negative number is not allowed in the mathematics of real numbers. If you change the initialization to “y:=4” you can get a solution again. However you will only be able to find the (3, 9) solution, not the (1, -1) solution, if you write the equations this way, since the numerical function “sqrt()” in Real Answer only accepts positive inputs and produces only positive results. The moral here is that is often advantageous to write the equations in forms showing a variable being multiplied by itself instead of having a square root taken. Self-multiplication is faster, more accurate, and can work on variables that have negative values.
