Following on from the under-whelming success of my recent SAND+SUN+SEX+SEA=IBIZA alphametic puzzle and its solution, I recalled another exercise in mathematics from earlier this year. One of my granddaughters sent me a text message asking if I could solve the following 3-variable set of linear simultaneous algebraic equations.

x+2y+3z = 14 (Eqn. 1)

2x+3y+4z = 20 (Eqn. 2)

5x+5y+5z = 30 (Eqn. 3)

Easy-peasy I thought. I enjoy algebra and this would be a doddle—the sort of thing you would have done in year 8 or 9. A couple of substitutions and out would pop the answer. So, I set about substituting, expanding and tidying up and singularly failed to solve the equations. Watch…

From Eqn. 1, x = 14 – 2y – 3z.

Substituting for x in Eqn. 2, 2(14 – 2y – 3z) +3y + 4z = 20 which tidies to y + 2z = 8 (Eqn. 4)

Substituting for x in Eqn. 3, 5(14 – 2y – 3z) +5y + 5z = 30 which tidies to, guess what, y + 2z = 8 (Eqn. 4 again)

Now what? I scratched my head, chewed the end of my pen, stared out of the window and generally waited for inspiration. I was particularly intrigued by the fact that there was a solution. Apparently the mathematics’ teacher at my granddaughter’s school had asked the pupils to derive the solution x = 1, y = 2, z = 3. This solution works. Try it.

Gabriel Cramer, Swiss mathematician, 1704 – 1752

Finally, I remembered from the dim distant past (the mid-60s in my case) that there was a plug-in-the-numbers-and-turn-the-handle method based on determinants. Jeez, I could barely spell determinants let alone recall what they were. I retrieved one of my degree-course books on mathematics and there it was—Cramer’s Rule. Yes! All I had to do was to remember how to apply it. That took a bit of deep memory retrieval, aided by my maths book and a very useful website, plus a few cups of coffee but I did it. Again, if you are interested to see the workings, send me an e-mail and I’ll send you my solution, no strings attached.

The bottom line is that applying Cramer’s Rule did not come up with a solution. For those who know about this rule, the denominator determinant, D, given by the coefficients of x, y and z computes to 0. This is bad news as D is a denominator in the final calculations—that is:

x = D_{X}/D

y = D_{Y}/D

z = D_{Z}/D

where D_{X }= the x-numerator determinant, D_{Y} = the y-numerator determinant, and D_{Z} = the z-numerator determinant.

In all cases, dividing by zero produces an undefined result. When this happens using Cramer’s Rule, the set of equations is termed inconsistent or indeterminate meaning the answer cannot be determined.

So, how did the maths teacher produce the x = 1, y = 2, z = 3 result? I asked my granddaughter to bring me home a copy of the teacher’s workings but, apparently, all he did was state the result without proof and then move on to another problem. Hmm. How did he do it? By trial and error? And why did Cramer’s Rule fail? To this day, I don’t know the answers to these questions.

By the way, a couple of comments on my alphametic posting suggested I am bored and have nothing better to do than work on mathematical problems which, by implication, is also boring. Nothing could be further from the truth. First, I am not bored. I have plenty of things to keep me occupied in my retirement. Second, I have never found mathematics to be boring—tough sometimes, perplexing other times but never boring. Third, the ex-engineer in me rises to the challenge of solving the problem. And fourth, the ex-teacher in me rises to the challenge of presenting a solution that is understandable even if mathematics is not your thing. It’s interesting that those who claim I am bored never asked for my solution. What can you infer from that?

(^_^)

Mary

said:I have studied your last two Blog’s with interest and have come to the conclusion that you are definitely a Nerd – and I mean that in the nicest possible way cousin!

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Ben Bennetts

said:Just to be clear, here’s the Oxford Dictionary’s definition of a nerd:

a person who lacks social skills or is boringly studious. I feel really bad that you think that about me. Did you mean geek:knowledgeable and obsessive enthusiast?I do hope so!LikeLike

Mozz

said:Reference your final comment maybe the inference is that they too are bored!!?

“Ben the Geek” – sounds about right!

Mozz

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Mary

said:I’m very sorry if I have upset you Ben and I do not think that you are lacking in social skills or boringly studious. I accept the Oxford Dictionary’s definition of ‘nerd’ but I still think it sounds more knowledgeable than a ‘geek’. Perhaps ‘nerdy’ has become one of those endearing words that is removed from the original definition. On the other hand perhaps I just watch too much BBT.

I would be very sad to lose any of my ‘nerdy’ relatives or friends as they make me laugh and brighten the world.

Live long and prosper – see I can also be a nerd/geek!!

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Ben Bennetts

said:Sticks and stones may break my bones but words will never hurt me… unless the word is nerd.

To call me a nerd is rather absurd

when clearly I’m chic and unique.

But I’m not deterred by the word I heard:

I have mental physique,

an air of mystique,

all of which makes me a geek!

Please excuse the unusual meter of this poem. It’s a geeky thing.

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Paul

said:Ben,

Wild guess: the three equations given are really just 2 equations (the 3rd is simply a modified form of one of the first two). This leads to an ill conditioned determinate in Cramer’s rule. So the school teacher had all three equations available but in effect just gave you two of them.

No you are not a geek!

Paul

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Ben Bennetts

said:Yes, I too suspected that one of the equations was a reinvention of another but it’s not. The {x, y, z} coefficient sets are all different: {1, 2, 3}, {2, 3, 4} and {5, 5, 5}. ‘Tis still a mystery.

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Paul

said:The 3 equations in 3 variables must be independent if the equations are

to be solved. Unfortunately the 3 equations given are not independent –

you can produce equation 3 by subtracting 1 from 2 and scaling by 5. I

think this means that there are an infinite number of solutions!

Now I can almost hear you saying “OK, give me another solution! “. Well

try x= 3, y= -2, z= 5 for example.

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Ben Bennetts

said:Excellent. Well done Paul. The mystery is solved and Cramer is off the hook.

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