Question

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Answer 1

would the set up be 2(6x+y)=6xy ?

Answer 2

Well, you actually have several pieces of info.

two positive numbers <- Need to keep that in mind.

twice their sum <- If you use x and y that is 2(x+y)

equals their product = xy

and one number is 6 times the other number. <- Now that lets you change from x and y to just x. See if 6x=y you can replace some stuff and get it in one variable.

Answer 3

wait I am not really understanding the 6 times the other number

Answer 4

im having difficulties writing the equation with that

Answer 5

oh so you're saying that I could just use x and have one that says 6x

Answer 6

Hmm... Yes, but it comes out oddly... I am checking it to see.

Answer 7

2(6x+x)=6x x

Answer 8

You're doing well. just keep going until you find one equals 14 and the other equals 7/3

Answer 9

Yes. Those are the answers, but because of the curve... hmm...

Answer 10

Ah, a partial substitution allows for finding it.

Answer 11

wait then what was your set up if you got 7/3 and 17..?

Answer 12

wasmy equation correct

Answer 13

See, a full substitution does:

2(x+y) = xy

2(x+6x) = x 6x

2(7x) = 7x

14x = 7x

And that is only true for x=0. They want positive integers and 0 is not positive so it is an extraneous answer.

But if you do this:

2(x+y) = xy

2(x+6x) = xy

You can solve for y. Then use y to solve for x.

Answer 14

I got 7/3 and 14; not 17

Answer 15

wait how did you get from x 6x to 7x

Answer 16

x.6x = 6x^2

Answer 17

oops. I was thinking +.... hmm. Odd that it still worked.

Answer 18

I knew my math was off somewhere. Was trying to figure out where. Had a mental slip and could not see where but felt it was wrong.

Answer 19

oh ok

Answer 20

I am getting 14 but i didnt get 7/3

Answer 21

14x = 6x^2

0 = 6x^2 - 14x

0 = 3x^2 - 7x

0 = x(3x - 7)

There is the 7/3

3x - 7 = 0

3x = 7

x = 7/3

Answer 22

So much easier when I fix the math. LOL.

If you look at it graphically you can see why the answers are a bit odd to find. It is not a simple line intersection:
https://www.desmos.com/calculator/cioobyvoc3

But in either case for the "one number is 6 times the other number" you get the same solutions. Just their order is different if viewed as an ordered pair.

Answer 23

oh ok!

Answer 24

Sorry about the \(x^2\) mistake. Anyhow, a partial subsitution finds the y valye directly.

2(x+y) = xy

2x+2y = xy

2x+2(6x) = xy

2x+12x = xy

14x = xy

14 = y

So that is one way to confirm in this case. The graph is also another nice check. Always good to do it a few ways so you can make sure of the answer.

Answer 25

ah okay tots makes sense thnx!

Answer 26

That's ok!. We are humans

Answer 27

Oh, and the 14x = xy is also true when x = 0. hehe. So it does work for both times this happens on the graph

Answer 28

And it is always the silly little mistakes that get us.