Math Is Fun Forum

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Another kind of linear equation I never saw

Hi everyone. I saw this problem in an exam of a junior class in my school (SHS 1 or [first year - high school]) anf shame on me,  I can't get around it, since last two terms.

My age is a multiple of seven. Next year it will be a multiple of five. I am more than 20 years but my age is less than 80. What is my age? I'm sure and I hope answer is posted with solution/procedure.

Many thanks in advance.

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Only a friend tells you your face is dirty.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Another kind of linear equation I never saw

Hi;

The problem easily responds to trial and error and that will be fastest.

age = 7 n

age + 1 = 5m

7 n = 5 m - 1

By inspection n =2 and m = 3 is a solution but is ruled out. We try each one because the number is small and get n = 7 and m = 10 meaning the age is 49.

There are at least 4 analytical means to arrive at the solution.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Another kind of linear equation I never saw

hi math9maniac

Let's say your age is A.

Then A = 7x where x is a whole number.

And A+1 = 5y where y is a whole number.

So 5y = 7x + 1

If I remove the restriction about whole numbers then solutions will all lie on the straight line 5y = 7x + 1.

So whole numbers solutions will occur if that line goes through any points where both coordinates are whole numbers.

If you can spot one answer eg (2,3) 5x3 = 7x2+1,  then you can find all the whole numbered solutions by adding 5 to the x coordinate and 7 to the y coordinate.
(can you see why this works?)

Thus (2+5, 3+7) is the next solution and (2+5+5,3+7+7) is the next.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: Another kind of linear equation I never saw

To solve the following Diophantine equation

[list=*]
[*]

[/*]
[/list]

Otherwise there is no solution. For example, 2x + 4y = 9 has no solution since the LHS is always even while the RHS is odd.

For example, 2x + 4y = 10  → x + 2y = 5. This step is optional but it helps by making the numbers you are working with smaller.

This can be done by trial and error. If the numbers involved are large, the Euclidean algorithm might be useful. (NB: A particular solution always exists. See following post.)

[list=*]
[*]

[/*]
[/list]

Proof: Check first that

[list=*]
[*]

[/*]
[/list]

Now suppose x=xʹ, y=yʹ is another solution. Then

[list=*]
[*]

[/*]
[/list]

QED.

Last edited by Olinguito (2015-04-05 21:15:02)

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Bassaricyon neblina

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: Another kind of linear equation I never saw

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Bassaricyon neblina