Fastest way to find consistency/solutions of equations

[Krushnaraj Pandya]

Homework Statement

Consider three equations in three variables, To check their consistency usually the adjoint of the matrix (from coefficients of x,y and z are found) and the inverse is found for the solution. I have heard of cramer's rule but it has a similar complexity.
These same kind of questions are repeated every year in the JEE which makes it very important for students like me to figure out something fast since there's no time to be calculating inverses and adjoints and risk mistakes as well.
As a sample- Here's a question from 2017 JEE Mains
Find the set of distinct values of b for which the following equations has no solution.
x+y+z=1; x+ay+z=1; ax+by+z=0

Homework Equations

Any pertaining to algebra

The Attempt at a Solution

I thought of some ways which might be faster but I don't know if they are foolproof-
1)Solving them pairwise- (easier for a solution- but I don't know how consistency can be checked)
2) Using 3d geometry, these are equations of planes. I'm not sure how exactly we can proceed using that though
3) As for the sample question, setting determinant=0 gives a=1 which means two of the equations are identical- x+y+z can't be 0 and 1 at the same time (I suppose but I'm not sure) so b is 1. Is that the right way to go about it?

 

[member 587159]

Homework Statement

Consider three equations in three variables, To check their consistency usually the adjoint of the matrix (from coefficients of x,y and z are found) and the inverse is found for the solution. I have heard of cramer's rule but it has a similar complexity.
These same kind of questions are repeated every year in the JEE which makes it very important for students like me to figure out something fast since there's no time to be calculating inverses and adjoints and risk mistakes as well.
As a sample- Here's a question from 2017 JEE Mains
Find the set of distinct values of b for which the following equations has no solution.
x+y+z=1; x+ay+z=1; ax+by+z=0

Homework Equations

Any pertaining to algebra

The Attempt at a Solution

I thought of some ways which might be faster but I don't know if they are foolproof-
1)Solving them pairwise- (easier for a solution- but I don't know how consistency can be checked)
2) Using 3d geometry, these are equations of planes. I'm not sure how exactly we can proceed using that though
3) As for the sample question, setting determinant=0 gives a=1 which means two of the equations are identical- x+y+z can't be 0 and 1 at the same time (I suppose but I'm not sure) so b is 1. Is that the right way to go about it?

I would go with the determinant approach.

Determinant = 0 gives the points where there is no unique solution of the system. It can still happen that you have infinitely many solutions in this case, so each of the problematic points have to be checked individually.

 

[StoneTemplePython]

can you do gaussian elimination? This would give you determinant as a by product, but also easily allow you to check for consistency if the matrix is singular (i.e. figuring out what is in the nullspace)... I would almost never recommend explicitly inverting a matrix for calculations, especially by hand.

I think for 3x3 matrices Cramers rule may technically be faster for non-singular matrices though...

 

[Ray Vickson]

Homework Statement

Consider three equations in three variables, To check their consistency usually the adjoint of the matrix (from coefficients of x,y and z are found) and the inverse is found for the solution. I have heard of cramer's rule but it has a similar complexity.
These same kind of questions are repeated every year in the JEE which makes it very important for students like me to figure out something fast since there's no time to be calculating inverses and adjoints and risk mistakes as well.
As a sample- Here's a question from 2017 JEE Mains
Find the set of distinct values of b for which the following equations has no solution.
x+y+z=1; x+ay+z=1; ax+by+z=0

Homework Equations

Any pertaining to algebra

The Attempt at a Solution

I thought of some ways which might be faster but I don't know if they are foolproof-
1)Solving them pairwise- (easier for a solution- but I don't know how consistency can be checked)
2) Using 3d geometry, these are equations of planes. I'm not sure how exactly we can proceed using that though
3) As for the sample question, setting determinant=0 gives a=1 which means two of the equations are identical- x+y+z can't be 0 and 1 at the same time (I suppose but I'm not sure) so b is 1. Is that the right way to go about it?

This question is in the wrong forum: it should be in "precalculus" mathematics. However, I will answer it anyway.

The standard way is to use Gaussian elimination. So, use the first equation to solve for x in terms of y and z. Substitute that expression for x into the second and third equations. Now the last two equations contain y and z only, so are easier to examine. If need be, you can even use the new second equation to solve for one of y and z in terms of the other. Then substitute that expression into the third equation, and you will be left with a single equation in a single variable.

Solving $(1): \; x+y+z=1$ for $x$ gives $(1a): \;x = 1 - y - z.$ Substituting (1a) into $(2): \; x + ay + z = 1$ gives $1-y-z+ay+z=1,$ which simplifies to $(2a): \; (a-1) y =0.$ Substituting (1a) into $(3):\; ax + by + z = 0$ gives $a -ay - az +by + z = 0,$ which simplifies to $(3a): \; (a-b) y +(a-1) z = a.$

So, our new equations are $\; x+y+z = 1, \: (a-1) y = 0, \: (a-b)y +(a-1)z = a.$ Now we can start to examine cases.
(a) If $a \neq 1$ the second equation forces $y = 0.$ Then, the third equation has solution $z = a/(a-1).$ We get $x$ from the first equation.
(b) If $a = 1$ the second equation allows $y = \text{anything}.$ You can then look further into consequences of the third equation (when you have a unique solution for $z$ or no solution at all for $z$).

Note that instead of solving and substituting, we can obtain the same results by adding multiples of one equation onto the others, to eliminate a variable. All of that can be sped up and mechanised by looking at the detached-coefficient forms. The original equations are
$$\begin{array}{ccc|c}
x & y & z & \text{RHS} \\ \hline
1&1&1&1\\
1&a&1&1\\
a&b&1&0
\end{array}$$
To eliminate $x$ from the last two rows we use row-operations: (New Row 2) = (old row 2) - (Row 1) and (New Row 3) = (Old Row 3) - a×(Row 1). That gives
$$
\begin{array}{ccc|c}
x & y & z & \text{RHS} \\ \hline
1&1&1&1\\
0& a-1 & 0 & 0 \\
0 & b-a & 1-a & -a
\end{array}$$

We can abbreviate the row operations as $R2 \leftarrow R2 - R1$ and $R3 \leftarrow R3 - a \times R1.$

All such row operations can be turned into matrix multiplications on the left, and that is essentially the way modern linear solving methods deal with such systems. Almost nobody uses determinants on large systems, unless they happen to have special structure. (In fact, the way most computer packages evaluate determinants is to first perform Gaussian elimination steps to obtain a so-called "LU decomposition", then compute the determinants of $L$ and $U$, both of which are very easy.)

 

[Mark44]

This question is in the wrong forum: it should be in "precalculus" mathematics.

I think I moved the thread before you posted.

 

[Ray Vickson]

I think I moved the thread before you posted.

Not on my system! All I saw before starting to compose my response was messages #1 and #2. Furthermore, my system did not show this thread moved to the pre-calculus forum. After I pressed the enter key, suddenly post #3 and your "move notice" showed up on my screen, along with my post, of course.

That type of thing happens to me a lot, so I often post almost-duplicate messages to others but which I had not seen before pressing the enter key, and my posts sometimes do not acknowledge the contributions of others, for the same reason.

 

[epenguin]

I don't know what JEE is, probably something entrance exam.

However I do not like the exam questions from your country (India I guess) which too often seem to be privileging an artificial smartassedness rather than the scientific understanding that your other helpers here are trying to give you. Indeed you mentioned the need for speed.

Maybe you are supposed to look at the first two equations and say right off, these must obviously be incompatible unless $a = 1$. Well I hope it is obvious now I have said it. Maybe you are then supposed to say The equations have no solution for all $b$ when $a ≠ 1$

In the case that $a = 1$ the system reduces to two equations

$$ x+y+z=1$$
$$ x+by+z=0$$
which obviously has no solutions when $b=1$.I will leave it to you and hope you will easily see that otherwise these two equations are compatible (have actually an infinity of solutions).

Of course I would have to withdraw my criticisms of your system in this case if it turns out you had mistranscribed the questions, which is a frequent cause of rubbish questions on this site.

 

[Delta2]

Just being curious, because you are asking too often for the most convenient/easiest/fastest way to solve something, how many questions are there in the JEE exams and how much is the total time of the exams?

And what JEE is, is it an Electric Engineering school? I have seen some problems that you posted that involve 3D geometry, strange that they check 3D geometry for electric engineers, I guess 3D geometry is needed for architects/civil engineers/mechanical engineers but for electric engineers its too much I think.

 

[Krushnaraj Pandya]

I think I moved the thread before you posted.

Not on my system! All I saw before starting to compose my response was messages #1 and #2. Furthermore, my system did not show this thread moved to the pre-calculus forum. After I pressed the enter key, suddenly post #3 and your "move notice" showed up on my screen, along with my post, of course.

That type of thing happens to me a lot, so I often post almost-duplicate messages to others but which I had not seen before pressing the enter key, and my posts sometimes do not acknowledge the contributions of others, for the same reason.

Sorry about that, I was half asleep and clicked on the physics forum out of habit...I am working through all the answers one by one and I really appreciate all the work you put into write them

 

[Krushnaraj Pandya]

I don't know what JEE is, probably something entrance exam.

However I do not like the exam questions from your country (India I guess) which too often seem to be privileging an artificial smartassedness rather than the scientific understanding that your other helpers here are trying to give you. Indeed you mentioned the need for speed.

Maybe you are supposed to look at the first two equations and say right off, these must obviously be incompatible unless $a = 1$. Well I hope it is obvious now I have said it. Maybe you are then supposed to say The equations have no solution for all $b$ when $a ≠ 1$

In the case that $a = 1$ the system reduces to two equations

$$ x+y+z=1$$
$$ x+by+z=0$$
which obviously has no solutions when $b=1$.I will leave it to you and hope you will easily see that otherwise these two equations are compatible (have actually an infinity of solutions).

Of course I would have to withdraw my criticisms of your system in this case if it turns out you had mistranscribed the questions, which is a frequent cause of rubbish questions on this site.

Just being curious, because you are asking too often for the most convenient/easiest/fastest way to solve something, how many questions are there in the JEE exams and how much is the total time of the exams?

And what JEE is, is it an Electric Engineering school? I have seen some problems that you posted that involve 3D geometry, strange that they check 3D geometry for electric engineers, I guess 3D geometry is needed for architects/civil engineers/mechanical engineers but for electric engineers its too much I think.

JEE stands for joint entrance examination, which is an entrance to all engineering/all branches across India. It relies solely on rote memorizing, blind practice of similar questions and conceptual clarity can go out the window. Some of my friends don't post here on PF just because they have so many questions to solve and they need quick answers like "just do this in these type of questions" and move on. It has reached the height of stupidity, while students who are 12 years old have started preparing for it. The trouble is 11,00,000 students appear for jee mains, 2,00,000 qualify that and appear for jee advanced, 10,000 clear it but realistically only 5,000 seats have some value in established IIT's. Here the students select branches based on their rank more often than their interest, since some branches have higher placements and salaries and IIT's are a tag that that person can work like hell. (the jobs aren't engineering related mostly, ironic).

Even more surprising to you will be that, we have to pay not to go to school (which are called dummy schools) so we can instead go to a coaching class all day- which is just another school but focused only on JEE- and they have gathered such a huge industry that you can see TV ads, youtube ads, front full page newspaper ads across the country for particular institutes. India's education system currently is like a scam but there isn't anything anyone can do to filter so many students since everyone wants to be an engineer/doctor, no wonder if the questions aren't good- I personally dislike the whole thing. I've been preparing for two years without enjoying a single weekend (you guessed it, coaching classes are full time) and if I don't do well I'll have to prepare for another year.

Coming to the need for convenient solutions, there are 90 objective type questions with +4 marks and -1 for an incorrect answer to be done in 3 hours. The number of questions a person goes with a target to attempt is never 90 though- In the history of jee only one student has had a perfect score and he was preparing for it since the 7th grade. So now the issue is to solve as many as possible- I'm in a fix right now since inside 3 hours I can do only 30 questions or so but given some more time I can go upto 65. These scores mean either a <1000 rank or not clearing the cut-toff- doesn't matter how clearly I've understood things. They don't divide things for different branches either- 3D geometry is in the exam so you have to do it whether you are an aspiring electrical engineer or not.

My jee mains is on 11th January so I'm in a bit of a time crunch, while feeling at the same time that a lot of my hard work is in vain just because I won't be able to memorize formulas or revise an old concept or not be able to do the majority of questions I know in the given time. Calculators aren't allowed for some reason. Jee advanced will be even more notorious, those papers have up to second year engineering questions and nobody has scored >75% in it. Those close to that score are a very very tiny number. There's no fixed pattern but generally It has questions with +4, -2 for incorrect. A multiple correct type mcq which if you mark 2 correct options but don't mark the 3rd correct one you get a -2. A matching type if you match 7 options correctly but don't mark one, you get a -2. Currently I have less time for jee mains though so I'm focusing just on that. Checking consistency is a fixed question every year, so are two 3d geometry questions, half the paper is actually fixed but I'll probably not be able to attempt it

 

[Delta2]

I see now, seems to me the JEE are a real pain in the ass. I understand now why you seek for the easiest and fastest method.

Here in Greece there are similar exams for entrance in the state universities (you can enter private universities/colleges without exams but you have to pay). But the exams questions are far less and usually organized in 4-5 themes (with many sub questions per each theme). Total time is 3hours also.