Question

The problem says:
Find all pair of integers such that when they are added they equal 100 and when they are multiplied they are at least 2000.

This is what I have so far:
x+y=100
xy=2000

Answer 1

the second part should be
xy >= 2000

Answer 2

I solved for x:
x=100-y
then i substitute x into the second equation

Answer 3

The expression "at least" is a key word that an inequality is being used.

Answer 4

Continue what you are doing, but use @cwrw238 's inequality to plug it into.

Answer 5

ok.. let me try that.

Answer 6

(100-y)y > 2000

Answer 7

ok except its greater than or equal to >=

Answer 8

\[(100-y)y \ge 2000\]

Answer 9

right

Answer 10

now froget the > for time being and solve the quadratic equation

Answer 11

100y-\[y^{2}\] \[\ge 2000\]

Answer 12

ok let me try to solve it

Answer 13

let me re-do
\[100y-y ^{2} \ge 2000\]

Then I get \[-y ^{2} \ge 2000-100\]

Answer 14

you need to solve
y^2 - 100y + 2000 = 0

Answer 15

this wont factor so you'll need to use quadratic formula

Answer 16

are you ok with doing that?

Answer 17

i'll be back in 3-4 minutes

Answer 18

\[x= \frac{ -(-100) \pm \sqrt{(-100)^{2}-4 (1)(2000)} }{ 2 }\]

\[x \frac{ 100 \pm \sqrt{1000- 8000} }{2}\]

Answer 19

\[x= \frac{ 100 \pm \sqrt{-7000} }{ 2 }\]

Answer 20

right
you'll find that comes to 72.36 and 27.64

Answer 21

ok did i miss a step because i dont get 72.36 and 27.64

Answer 22

100^2 = 10,000

Answer 23

error there

Answer 24

oh i see! thank you so much. I was trying to factor it. Now i now why i wasnt getting it.

Answer 25

so we see one pair of integers already
72 and 18 = they add up to 100 and 28*72 = 2016

Answer 26

right! yes, i see it now. and since it says at least 2000 its ok to get 2016 right

Answer 27

once i go back to check the answer i get 100 and 2016

Answer 28

yes

Answer 29

but thats not all the possible pairs
eg 71 and 29 gives 100 and 2019

Answer 30

I'm not quite sure how to continue form here....

Answer 31

thats more than enough. Thank you so much!

Answer 32

ok - no problem

Answer 33

I got the graph on my calculator
pairs of numbers which add up to 100
which are along the heavy line fit the bill