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Problem

Suppose and are integers. How many solutions are there to the equation \(ab = 2a + 3b?\)

HelpPls Jul 4, 2018

#1**+2 **

a b = 2 a + 3 b

a = (3 b)/(b - 2), and b cannot=2

b = (2 a)/(a - 3), and a cannot=3

Solutions:

a = -3, b = 1

a = 0, b = 0

a = 1, b = -1

a = 2, b = -4

a = 4, b = 8

a = 5, b = 5

a = 6, b = 4

a = 9, b = 3

Guest Jul 4, 2018

#2**+1 **

Thank very much but your 8 solutions were different with the answer key... Can there be differernt 8 solutions?

HelpPls
Jul 4, 2018

#3**+1 **

I am very please with your response HelpPls.

A thank you and a follow up question (for our guest answerer)

There can be an infinite number of answers because there are 2 unknowns (a and b) and only 1 equation.

I have graphed it here for you.

Every point on the graph is an answer but most are not integer answers so they are no good to you.

I have plotted all of our guests answers, there may be many more, I have not thought about it well enough to say.

If you plot each of these points into the equation it should make it true (i have not checked them all but they all seem to sit on the graph nicely.

\(ab = 2a + 3b\)

for instance take the point (4,8) a=4 and b=8

ab=4*8=32

2a+3b= 2*4+3*8 = 8+24 = 32

left hand side = right hand side so it is one of the answers. :)

Melody
Jul 4, 2018

#4**0 **

You can ask more questions if you want but keep in mind that sometimes answerers don't see them.

You can send private messages to members and send a link to the question if you want to make sure they see it.

( they may not be on often though)

Maybe another person will step in and help you though :))

Melody
Jul 4, 2018