Question

Find all integer solutions to 14x + 77y = 70

Answer 1

7(2x + 11y) = 70
2x + 11y = 10
lol - i'm lost now
if there are solutions then 1 must be negative

Answer 2

It's better to solve 2x + 11y = 1 first, then scale both sides appropriately.

Answer 3

Notice how 11 = 2*5 + 1, so


11 = 2*5 + 1

11*1 = 2*5 + 1

11*1 - 2*5 = 1

2(-5) + 11(1) = 1


This means that the solutions for 2x+11y = 1 are x = -5 and y = 1

Answer 4

Once you know that

2(-5) + 11(1) = 1

multiply both sides by 10

2(-5) + 11(1) = 1

10( 2(-5) + 11(1)) = 1*10

10*2*(-5) + 10*11*(1) = 10

2*(-5*10) + 11*(10*1) = 10

2*(-50) + 11*(10) = 10


So the solutions to 2x+11y = 10 are x = -50 and y = 10

Answer 5

That's just one integer solution, but you can use it to find the rest.

Answer 6

@jim_thompson5910 Why did u nt take 10 at RHS.
Plz tell:)

Answer 7

because it's much easier to solve the equation when the RHS is 1, then you scale appropriately

Answer 8

oh i see.
thnx:)

Answer 9

It's shown in the work above

Answer 10

clever stuff!

Answer 11

of course:)
Previous was much clever:)

Answer 12

\[\begin{align}
14x+77y&=70\\
\therefore2x+11y&=10\\
\therefore11y&=10-2x=2(5-x)\\
\therefore y&=\frac{2(5-x)}{11}\implies5-x=11n\text{ (i.e. (5-x) must be a multiple of 11)}\\
\therefore y&=\frac{2*11n}{11}=2n
\end{align}\]so solutions are:\[x=5-11n\]and\[y=2n\]for:\[n=-\infty\text{ to }+\infty\]

Answer 13

Here's something that our tutor thought was the right answer.
x=-50+11k
y=10-2k
for all k integers

Answer 14

there are many forms for the /correct/ answer. they all generate the same pair of x and y values

Answer 15

e.g. this is what wolfram comes up with: http://www.wolframalpha.com/input/?i=solve+14x+%2B+77y+%3D+70+over+integers

Answer 16

wolfram's solution uses an 'n' that is the negative of my 'n'

Answer 17

ok, back to the other question...with a negative sign, are you familiar with the continued fraction method?

Answer 18

lets go back there...