Systems With 3 Variables Worksheet Answers

[Jenifer Griffard]

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The worksheets enlisted in this post deal with a system of linear equations involving 2 or more variables. There are also word problems that need to be solved after framing a system of linear equations represented by each.

So I'm struggling to understand exactly what is the issue that is occurring that is stopping the system of equations below from solving. I know that it is solvable, I did it by hand, and I have the answers for verification. However, it is in the software claiming to be unsolvable. Am I entering it in improperly?

I was curious and tried the old symbolic engine (muPad) which was included up to version Prime 6. Not surprisingly its a bit more capable than the new one. It still throws a strange error initially ("Cannot understand the syntax") but we get results by either removing the units OR removing the roots:

My guess is that Prime's Symbolic is also unable to cope with the units assigned as it does not know anything about units. If you want to use the symbolic "solve", then better without numeric values at all, as shown by Luc.
But in the concrete case no symbolic solution is needed at all. A numeric solution block will do as well:

Okay, so I would like to ask if I understand this correctly: Due to limits with the design of prime with how it interacts with units, the original way that I was trying to solve cannot be done? If so, okay, that's something I can just adjust for.

As far as the solutions given, I actually tried the solve block method prior. However, when I did so, I would get this error where the solution from the solve block could be seen by evaluating it, as you shown in your example. However, when I would try to use it later in a symbolic solver, it would not register the variable as being initialized. I can simply use block solves to get around this issue, but they take up a ton of space on the document. Is there any way of getting around this, or is this just a hard limit on the software?

I may be wrong but it looks to me that Primes symbolics is not able to cope with your simple equations when the other variables already have assigned quantities with units. So you either use the symbolic solver before you assign values or you may clear.sym(...) to make the assigned values unavailable for the symbolic engine and add them later again to get a numeric result.

According using a solve block result in later symbolic evaluations: On contrary to real Mathcad, Prime has lost the ability to symbolically evaluate a solve block. Thats the reason symbolic caculations using numeric result from a solve block fail.

You may trick Prime by writing something like L.r:=L.r= after the solve block. You redefine L.r as being L.r and follow the assignment by a numeric evaluation. This stops the symbolic from trying to evaluate the solve block.
Here is simple example:

So now I'm just straight up confused. When I opened your document, it loaded and worked perfectly. I literally copied the entirety of that symbolic solve without the assume modifier into my document, and regardless, I'm getting the factor error. It however, works perfectly in your document.

All I could think of is that you copied the expressions at a position where one or both variables which are solve for had been assigned a value before. If thats the case you may try clear.sym(L.r,C.r) in front of the symbolic evaluation. But this could also be seen explained in the sheet I attached and so I guess its not the cause of the problem.

P.S.: My sheet was created in Prime 6 but using the new engine which is the same as in Prime 7 and 8 but it may have been improved/modified. Using Prime 6 enabled me to also test how the old engine (muPad) performs, but the sheet I attached was set up using the new engine. Does this sheet work OK, too, if you let it recalculate (F5)?

It's the exact same error that I was originally seeing in the OP. The second image is from your sheet, where it's showing the same error, so I believe that the problem with that and whatever is going on stopping me in my worksheet is the same issue. The first image in your worksheet works as you shown however. As far as why it works in yours, but not in mine, I have zero idea. Maybe it's using prime 6 legacy code on yours?

If you can you may post your worksheet and I'll have a look. But I can only "convert" it to Prime 6 format and try to open it there. The symbolic engine is the same in P6 and later versions, but it sure may be that they "improved" that engine and in doing so did brake things which were working in P6.

But then, if my worksheet is working OK when you open it in P8, then I guess that the problem is not Prime 8 but rather something which you may have defined in front of the position you inserted the expression OR it also may be that Primes auto-labelling bug has struck once again. Maybe the labelling of some variables was changed in the process of copying. Check, if ALL occurrences of L.r and C.r are still labelled "Variable" and not "-" (check also the variables after "solve").

What I see in your picture is that the unit kHz was not cancelled and still appears in the result. This should not happen! Obviously f.r is not replaced by the value you assigned to it. Did you assign a value (in kHz) in your new sheet?

While I tend to second Luc's advice, I also realize that this may mean that you end up with a lot of functions with ahuge number of arguments which may make expressions quite unclear and unreadable. So it depends.
If numeric results are all you want I still suggest using numeric methods like the "root" function or a solve block. Unfortunately solving systems of equations is only possible to do with the ugly looking and space consuming solve blocks and providing suitable guess values can be demanding for some systems.

You equations really aren't too complicated and I don't understand why Primes symbolics has that much troubles dealing with them (even without f.r being defined it should be able to provide a symbolic result). But the capability of the symbolic engine is quite limited and its very likely that sooner or later you end up with equations (maybe of higher order or where the variable is inside a log function and outside, too) the symbolic isn't able to solve symbolically and you are forced to numeric methods anyway.

So I actually figured out what was stopping my setup! I did not realize it, but I forgot that I renamed Fr above in the sheet, and that is what was causing the problems. Once I fixed that, it solved. I tried your method as well, and it had the same issue when I had it incorrectly labeled, but when I fixed it, it solved all the same.

John received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. John invested $4,000 more in municipal funds than in municipal bonds. He earned $670 in interest the first year. How much did John invest in each type of fund?

Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visual gymnastics.

In order to solve systems of equations in three variables, known as three-by-three systems, the primary goal is to eliminate one variable at a time to achieve back-substitution. A solution to a system of three equations in three variables [latex]\left(x,y,z\right),\text[/latex] is called an ordered triple.

Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.

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