Systems of Equations: What's your favorite way to solve them?

1. Jul 31, 2015

JR Sauerland

As the title suggests, I want the tips from the pros. I've learned all of the methods, but going into Calculus/Physics, which methods should I take preference to? Furthermore, will there be significant use of systems of equations in Physics or Calculus in college?
P.S.: I know how to graph the systems on my calculator so I usually 'cheat' so to speak that way and have my TI84 calc out the intersect. However, this is probably a bad and lazy habit as I've heard there are no graphic calculators allowed in Calculus

2. Jul 31, 2015

Dr. Courtney

I like the Matlab "solve" command.

3. Jul 31, 2015

SteamKing

Staff Emeritus
Most of the linear systems covered in math courses can be solved by hand, typically using Cramer's Rule for 2x2 or 3x3 systems. Anything larger will usually require the use of a computer, since the amount of calculation for elimination methods grows approximately as n3, where n is the number of unknown variables, and the probability of making a mistake in arithmetic also increases.

There is no "one best" method in my experience. That's why you are taught several different methods. You can examine a given problem and decide which method to use to provide the least amount of work to obtain a solution.

4. Jul 31, 2015

jack476

Systems won't really start to appear in full until the advanced courses. You might get some matrix algebra in a basic circuit theory course, but I think that's mostly it. Calculus 1-3 did not involve any systems, diff-eqs touched on them briefly, and linear algebra was where it really took off. So not in the first few semesters, but as you get into more advanced courses, especially in physics and engineering, most of your problems will involve dealing with systems. At the same time, and this may be a little more true for physics than engineering, you start to become more concerned with analyzing systems rather than solving them explicitly.

To cite an example in classical mechanics, an analytical solution of the three-body problem's system of differential equations is impossible. Instead, you'll be given mathematical tools to analyze this system and make reasonable predictions about it. You may have numerical packages to perform differential and integral operations arithmetically, there are also techniques like perturbation and variational methods and stability analysis you'll learn to apply.

In real-world problems the systems you work with may very well be vast and true solutions may be impossible, and in such cases you have tools like MATLAB to help you. But you need to know the theory behind those tools, and there will be situations where you need to design your own.

Graphing calculators are typically neither allowed nor even necessary in most math classes in college. Everything you need to do, you'll be taught to do either by hand or by using the recommended software package (ie MATLAB), the calculator simply will not help for the overwhelming majority of problems you'll encounter. I use my TI-89 as a Gameboy emulator far more than I use it for classes. What I recommend getting is a TI-36x Pro scientific calculator, since it's much faster to use for quick calculations and save yourself some wrist ache on homework problems. Or a good slide rule and impress some of your older professors :P

5. Aug 1, 2015

Dr. Courtney

This may be true at some schools, but it certainly is not true at the schools where I have taught and attended. Consider a two dimensional collision there the objects stick together and one needs to solve for the final velocity. This is a system of two equations and two unknowns. If a two dimensional collision is elastic, there are three equations (two momentum equations and the energy equation) and three unknowns. Consider an Atwood machine type problem. These yield systems, usually two or three equations in the same number of unknowns. Usually the unknowns are the tension(s) and the acceleration. In Calc 3, a common case of multiple equations and multiple unknowns is optimization problems. These often have 3-5 equations and the same number of unknowns. You have more unknowns if you introduce Lagrange multipliers to solve them. Even in Calc 1, optimization problems often have two equations and two unknowns (the objective function differentiated and set equal to zero, and the constraint function, usually where a function is equal to a constant value.) I can even recall a few introductory chemistry problems that require solving two equations and two unknowns.

Again, it depends where you are. I recall being happy enough for students to solve systems of equations on their TI-89s, as long as their work showed the set-up clearly (These are the three equations and three unknowns ...) as well as documented using the TI-89 to solve it. The TI-89 fell by the wayside in courses that allowed MMa or Matlab. If students didn't know how to solve systems in 2-3 unknowns by Calculus 3, I wasn't inclined to take class time to teach them. I knew that Pre-Calc, Calc 1, Calc 2, and both 1st and 2nd semester intro Physics had all spent time teaching this in different contexts.

6. Aug 1, 2015

JR Sauerland

Careful with what you say. EVERYTHING I learned in my college statistics class (at a public state college) was done through the calculator.

Also, I never used Matlab... Which is why I ignored your previous comment. My college used Mylabsplus, which is wholly different and does not have any 'solve' function or whatever it is. No idea what that even is...

7. Aug 1, 2015

JR Sauerland

P.S: Will I need to know about Matrices? One of the universities I plan on attending has 'omit' written across them on the syllabus, omit typically meaning it's taken out. I guess to clarify, I am going to be a science major, either Geology or Env-Science. We delve into either Life Sciences Calculus or Engineering Calculus (our choice), but in either, I've noticed Matrices aren't part of it. Included below is my Life Sciences Calculus 1 syllabus.... We get a choice of Life Sciences or Engineering Calc, but I don't see any reason why I would take Engineering.

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8. Aug 1, 2015

Dr. Courtney

Good point. I had forgotten about how much the graphing calculators can get used in a stats class. I tended to focus on using spreadsheets in my stats classes, but some use the calculators.

There are a number of mature software programs that work wonders with a solve command or something equivalent to it.

The easiest way to see the power of these might be Wolfram Alpha:

http://www.wolframalpha.com/input/?i=solve+3x+5y+3z=0,+4x-3y-12z=1,+-2x-4y+3z=4

Try a few on your own.

9. Aug 1, 2015

Dr. Courtney

I recommend that every scientist and engineer's mathematics education include matrices. Topics are often omitted in a Calculus curriculum for time reasons based on the reckoning they they have been or will be covered in other courses, not because they are not needed. Strictly speaking, matrices belong to algebra rather than to calculus, so when trimming a 5 hour engineering calculus curriculum down to a 3-4 credit hour calculus for the life sciences, matrices may be omitted. This decision is informed by the fact that most high school Algebra 2 and Pre-Calculus curricula (at least in the US) include a good bit on matrices, including going back and forth between linear equations in 2 and 3 unknowns and the equivalent matrix representations, as well as the ability to complete pencil and paper solutions to 2x2 and 3x3 matrix problems.

There are problems in every area of science that can be more easily modelled with matrices than without them.

10. Aug 1, 2015

JR Sauerland

Not meaning to correct you or challenge you but here in the state of Florida, I've been to three high schools (long story) and taken Algebra at all 3. I had trouble with math in High School. However, I have never, ever, encountered matrices, which is why I ask. I took to the internet to learn the things we were never taught in Algebra.
What isn't taught in Algebra 1 (10th grade), Algebra 2 (11-12th grade), Intermediate Algebra (Freshman college), or even College Algebra (Freshman college semester 2):
• Matrix/Matrices
• Conic Sections
<<Syllabus for College Algebra <<Syllabus for Intermediate algebra.

11. Aug 1, 2015

SteamKing

Staff Emeritus
There's algebra, and then there's algebra. In HS, there's a lot of math-free math courses being taught, and this also continues into college courses as well.

In my personal experience, my algebra classes (starting in 8th grade, mind you) hit linear systems early, with 2 equations in 2 unknowns. The next year, 3 equations in 3 unknowns were handled, along with learning the basics of matrix arithmetic, calculating determinants, learning elimination, and what not. This is why I find this discussion (and others) of leaving basic linear algebra to college quite puzzling. To me, it's like teaching addition in grammar school, but not tackling multiplication until freshman year at college.

I also find teaching and/or learning a math subject by manipulating the buttons on a calculator to be alarming and disheartening at once. What do you do if you lose the calculator, or it quits working? Your understanding of a subject should never be tied that intimately to one device. The calculator should be an aid to crunching the numbers, nothing else, IMO.

12. Aug 1, 2015

Dr. Courtney

Yeah, we are seeing the dumbing down of the math curriculum. It has been a few years since I taught high school math in public schools. Back in 2002-2007, matrices were in every Algebra 2 book I looked at and in every state curriculum I reviewed. Some Algebra 1 texts and curricula had matrices also.

With the advent of Common Core, vectors and matrices have been removed from the Algebra 2 and relegated to 4th year math, which means Pre-Calc for most college bound students. The on-line ALEKS system still has matrices in its Algebra 2, but most states have accepted the reduction in content and rigor represented by the Common Core. Too bad.

Thanks for bringing this to my attention.

13. Aug 1, 2015

Dr. Courtney

I agree. But after a few weeks of having students solve 2x2 and 3x3 systems by hand, I like to allow the use of calculators/computer programs so that the students can focus more time and energy on word problems and applications rather than on additional weeks of solving them by hand every time. Technology also opens the door to 4x4 and larger which are prohibitively difficult to solve by hand if one hopes to complete a reasonable number of problems in the hour or two available in a given day.

14. Aug 1, 2015

JR Sauerland

I understand the logic of your point about the calculator not working or being lost. But seriously, think about the inverse of your logic for a moment (not criticizing you, just saying)... What do you do if you're in the field, and you need to calculate some crazy physics formula, and you don't have a pen and paper, but instead a calculator? What if you're in space, orbiting around Earth in the space station, and you don't have a pen and a piece of paper even? Are you really going to be able to create several 5 by 5 matrices and multiply them by each other, all in your head? Are you really going to be able to differentiate and do integrals (I say do because I have no understanding of Calculus because I haven't taken it) without a pen and paper? Furthermore, after being out of college for 5+ years and being an engineer or physicist, are you going to be top notch at solving those matrices and doing all of the hardcore math when you've been using a calculator for 5+ years? I think not having a calculator is worse in most practical scenarios than relying on one, just to be perfectly honest.

However, if we're talking about getting the concepts down, I completely understand the reasons behind why a student shouldn't have a calculator to do all of the work for them. If they never learn it by hand, how can they know what they're doing?

Uhhh, you're welcome I guess? :P

This is exactly what I'm getting at. Sure, it's great to do it by hand, but it takes an incredible amount of time, and you can also make mistakes very easily by solving matrices by hand. I know this because I am learning them now, and I have slight dyslexia, so it is a little bit unfair. However, a calculator does not have dyslexia, and will compute the matrices and the calculations at an exponentially faster rate than my mind and my hands writing them on paper can. P.S.: To those that will say 'if you have dyslexia you can still make mistakes on the calculator and mix up the numbers.' Blah blah blah, heard it all before. Yes, I know this. The key to getting the problem right is to double check the work you're doing. Before hitting the enter button, I go through and check over what I'm inputting and compare it to what has to be done.

15. Aug 1, 2015

Dr. Courtney

Another reason why it is important to teach pencil and paper methods first is to help students gain the experience needed to spot errors in computer/calculator generated results.

Competent students should learn how to solve 2x2 and 3x3 systems by hand before being allowed to use technology so they develop intuition and experience needed to double check and apply technology generated solutions, and so they can know what technology is doing when solving larger systems. I've used technology to solve systems 10,000 x 10,000 and larger. That is awesome, but using that technology adeptly really requires the experience of solving smaller systems with pencil and paper.

If it takes "an incredible amount of time" for a student to solve 2x2 or 3x3 systems, the student should not be seriously considering a STEM major until he gets his remedial math issues sorted out.

16. Aug 1, 2015

SteamKing

Staff Emeritus
I don't think this scenario is quite realistic.

Doing some simple mental arithmetic is a matter of training and discipline. A lot of people get confused when presented with 2 + 2 = ? and they don't have a calculator at hand.
I think I could still navigate my way around some simple integration and differentiation problems. That's what I do a lot here at PF.
I've been out of college 35 years, and I can still navigate without help, thank you very much. I still remember what I was taught about solving systems even further back than that, when calculators had just started to appear at prices which the typical student could afford.
Maybe I misinterpreted what you meant by this statement. We have had posters at PF who wanted to know, step-by-step, which calculator buttons to push to solve a problem, without getting bogged down in the messy details of actually understanding what they were doing.

I agree that using a calculator to solve a statistics problem comes in handy. I'm not a big fan of the "learning only which buttons to push" method of solving problems, though.

17. Aug 1, 2015

JR Sauerland

Yeah, but I'm not talking simple problems. I'm talking about Engineers working on serious issues. The foundations of buildings, sending a rocket to the moon, sending a ship to Mars. Human error is much more likely when problems are done by hand than they are calculated. If you accidentally make a mistake on paper, the ship full of humans could instead go barreling straight towards Jupiter

18. Aug 1, 2015

Dr. Courtney

This is not my experience, either as a teacher or as a scientist. Technology and computer programs all have bugs and are best applied by students and scientists who have considerable skill with pencil and paper calculations to empower double and triple checking the output of technology and computer programs.

Even in the above example where I used the Wolfram Alpha solve command to solve a 3x3 system, I would not recommend trusting the answer enough for a homework set solution without double checking it by hand.

Last edited: Aug 1, 2015
19. Aug 1, 2015

JR Sauerland

Applications like Wolfram Alpha and Mathlab are cheap applications slapped together with poor coding so that they are meager at best. Why I discredit them so highly is that they are not engineered or created by trusted companies such as Texas Instruments. Think about it. Do you think NASA entrusts Wolfram Alpha or the same technology as Mathlab? Absolutely not. But the applications NASA and other government entities use costs millions. That is the difference between the 'bugs' you speak of and actual reputable technology.

A calculator, on the other hand, is built off of a foundation of nearly a century of tested and proven science and engineering. Calculators seldom have bugs. Typically, it is the user using the calculator improperly that is the problem.

20. Aug 1, 2015

SteamKing

Staff Emeritus
I bought a cheap TI solar calculator one time. I didn't notice it until I took it out of the package, but it had two $\div$ keys but no × key. It was kinda kitschy, so I kept it, after I figured out which $\div$ key multiplied, and which worked as advertised.

I'm sure the people at Intel felt they had science and engineering on their side when the released the first Pentium processors, but alas, a nasty bug snuck into the circuitry, nevertheless:

https://en.wikipedia.org/wiki/Pentium_FDIV_bug

Most of the applications used by government which cost millions were probably programmed long ago before software became the commodity it is today. In fact, the legacy systems which exist nowhere else besides government are also a trap which serves to slow, or even discourage, modernization.

For example, the air traffic control system run by the FAA in the US was supposed to have been modernized decades ago, but a new system won't be in place and fully functioning until the next decade sometime:

https://en.wikipedia.org/wiki/Next_Generation_Air_Transportation_System

Your car probably has a more modern guidance system in its GPS navigation than the air traffic control system which governs take offs and landings at the local airport.

The point is, we rely on software to do many daily tasks, but we really don't know if it has been tested adequately to ensure correct design and function. Some systems are so complex, it is unlikely that they could be tested thoroughly, at least, not before they became obsolete. We still get software patches for the latest software dumped onto the internet every Tuesday, and this state of affairs will likely continue for the foreseeable future.

Unlike many professions, software writing is still often treated more like a cottage industry. Somebody develops a cool, new App, releases it, and makes millions of dollars before lunch. Does the software writer know how to count to 20 without using his fingers and toes? Who knows?​

Last edited: Aug 1, 2015
21. Aug 2, 2015

jack476

My mistake, then. I thought "calculator neutral" policies were the norm. All of my math classes were designed to not need a graphing calculator, it was part of our department policy. I think it was because they didn't want to deal with the possibility of students cheating by programming their calculators or the possibility that students might all be using different software packages that they found in some random repository on the internet.

22. Aug 2, 2015

SteamKing

Staff Emeritus
There are some settings where only approved devices may be used, usually in a test-taking venue.

For example, the SAT exam has this policy on which types of calculators are approved for use during the test:

https://sat.collegeboard.org/register/calculator-policy

Other standardized testing organizations, like those governing professional engineers, have similar policies.

It's best to check before signing up for a professional exam or college entrance exam to see what devices are permitted.

23. Aug 2, 2015

micromass

I don't get this. WolframAlpha is one of the most useful things in existence. I barely need anything else, certainly no calculator.

24. Aug 2, 2015

micromass

Sure, if you meet an actual system of equations in real life, then you will do it by computer or calculator. But by then you already know how to solve systems of equations. You must first learn how to solve things by hand, and only then can you rely to a computational device! In one of your other threads, you calculated $(-1)^{10}$ by calculator! This indicates a severe lack of understand which comes from using calculators too often.

25. Aug 2, 2015

MrAnchovy

Do you have any evidence for this, or is it a personal opinion?