Question

Compute the number of ordered pairs of positive integers (x,y) such that 3x + 5y = 500

Hint: you do not want to list them all, but you want to find a creative way to count how many first-quadrant ordered pairs satisfy this equation.

Answer 1

i know it says creative way but...

Answer 2

how do I find them ALLL??

Answer 3

is it like, find the x and y intercepts and then the slope and then just find your way from there??

Answer 4

what happens when x = 0 ?

Answer 5

y = 100

Answer 6

yes

Answer 7

when y is 0, x is ...

Answer 8

so (0,100) is one solution BUT x is not positive here

Answer 9

oh

Answer 10

what is the slope of this line?

Answer 11

um
5y= -3x+500
y= -3/5x + 100

Answer 12

so slope is -3/5x

Answer 13

just -3/5
it's the number out front of the x

Answer 14

oh wait, yeah, oops

Answer 15

slope = -3/5 = rise/run = (change in y)/(change in x)


what this means is that when x increases by 5, y decreases by 3


so (0,100) would turn into (5,97) following the rule above
notice how 0 turned into 5 (x increased by 5)
notice how 100 turned into 97 (y decreased by 3)

Answer 16

mmmhmmm

Answer 17

what about..

Answer 18

the question is: when does y become negative?

Answer 19

when... 33 ordered pairs later..?

Answer 20

but during that, x becomes negative

Answer 21

more like 34

100 - 34*3 = 100 - 102 = -2

Answer 22

oh I see what you mean

Answer 23

oh, yes

Answer 24

so would
so since x becomes negative faster than x, do i do something like 100-5x or something?

Answer 25

agh wait, i meant

Answer 26

so 100- 20x5...?

Answer 27

and then taking out the intercepts would mean I have 18 ordered pairs

Answer 28

to generate any ordered pair (x,y) where you want this point to be in Q1, use this rule

x = 0 + 5*k
y = 100 - 3*k

k is an integer
k ranges from k = 1 to k = 33

if k < 1 or k > 33, then (x,y) will not be in Q1 (it will be on an axis or in another quadrant)

Answer 29

so this shows that there are 33 ordered pairs (x,y) where x and y are positive integers

Answer 30

ohwait but what about the x being negative?

Answer 31

that happens when k < 1 which I mentioned above

Answer 32

oh, ok

Answer 33

thank you (like usual)

Answer 34

np