Question

find all pairs of integers whose difference is 5 and product is 234. So, what I got is -18 and 13. If i am right how do i go back to check if my answer is right.

Answer 1

\[x-y=5 \]
\[xy=234\]
If i go back to check for example:\[-18\times13=-234\]

Answer 2

if you answer is right
then the difference of -18 and 13 is 5 (is that a true statement?
And also the product of -18 and 13 is 234 (is that a true statement?

Answer 3

both of those statements have to be true.

Answer 4

are they

Answer 5

the difference of -18 and 13 is not 5 nor the product of -18X13 is not 234. They both give me a negative result

Answer 6

so that means that something went wrong when finding the other number
like I believe you got this far
(y+18)(y-13)=0
where y=-18 or y=13
right
but x=5+y

Answer 7

so if y=-18 then x=5+y=?

Answer 8

and if x=13 then x=5+y=?

Answer 9

right, i factored it out and thats how i got -18 and 13. so when I go back to check I use the x=5+y equation. So i said x=5+(-18) which x=-13

Answer 10

ok so (-13,-18) is one pair

Answer 11

what about the other one?

Answer 12

i made a type-0

Answer 13

well actually there are 2 sets of solutions

the numbers are -18 and -13

as well as the positive 18 and 13

Answer 14

and if y=13 then x=5+y=?
*

Answer 15

so that means that something went wrong when finding the other number
like I believe you got this far
(y+18)(y-13)=0
where y=-18 or y=13
right
but x=5+y


remember we had two y's so we will have two different x's

Answer 16

therefore two different pairs

Answer 17

ohhh i see my mistake I only solved for y. I didnt solve for x!

Answer 18

well you solved for y which we got was y=-18 or y=13
and we know x=5+y
so you already plugged in y=-18 giving you x=5+(-18)=-13

but you also need to plug in the other y which was y=13
into x=5+y

Answer 19

so x=5+13=18
So y=-18 and 13
and x=18 and -13

Answer 20

well i would list it as actually pairs
18,13 is one pair
-18,-13 is another pair

Answer 21

ah I see, got it! Thank you so much!

Answer 22

if you want list them in order
18,13
-13,-18

Answer 23

I put the bigger number first in each case

Answer 24

so we can have left number-right number=5 in each case

Answer 25

makes a lot sense now and what was my mistake.

Answer 26

what do you mean

Answer 27

i meant i get what was my mistake. I only solved for y and i didnt find ALL integers to give me the difference of 5 and the product of two integers to give me 234

Answer 28

in a way i guess you can say it wasn't really wrong it was just incomplete
but also -18 and 13 isn't a pair together that satisfies the difference being 5 and the product being 234.

Answer 29

right! thank you!