Set problem of x-intercepts and y-intercepts

[Lilt]

Homework Statement

Let S be the set of all lines with equation y = mx+b for which m+b = 36. For how many of the elements of S are both the x- and y-intercepts integers?

Homework Equations

y = mx+b for which m+b = 36

The Attempt at a Solution

I'm just not wrapping my mind around the steps in the solution here:

Denote by (a,0); (0, b) correspondingly x-intercept and y-intercept
of a line. Adding the equalities given in the problem and simplifying the result,
we will come to the equation of the set of the lines y = (m-1)x+36​

From that point onward, I can see and work through the steps. If I were to add these two equations, then I get y = mx + m + 2b - 36. I figure there's substitution occurring for their simplification, but I'm oblivious to it.

edit: Oh jeez, and I posted this in the wrong subsection. Should I re-post it, or will this be moved? Sorry about that!

 

[Villyer]

I found success by finding the x intercept of y=mx+b, and then analyzing what is necessary to make that intercept an integer.

 

[azizlwl]

1.

I'm just not wrapping my mind around the steps in the solution here:

Denote by (a,0); (0, b) correspondingly x-intercept and y-intercept
of a line. Adding the equalities given in the problem and simplifying the result,
we will come to the equation of the set of the lines y = (m-1)x+36​

It means the set of lines are all the lines that intercept y-axis at 36 and have gradient of (m-1).
But you data show y=(36-b)x+b
Lines intercept at y=b and have gradient of (36-b). So we have infinite numbers of solution.

 

[SammyS]

Homework Statement

Let S be the set of all lines with equation y = mx+b for which m+b = 36. For how many of the elements of S are both the x- and y-intercepts integers?

Homework Equations

y = mx+b for which m+b = 36

The Attempt at a Solution

I'm just not wrapping my mind around the steps in the solution here:

Denote by (a,0); (0, b) correspondingly x-intercept and y-intercept
of a line. Adding the equalities given in the problem and simplifying the result,
we will come to the equation of the set of the lines y = (m-1)x+36​

From that point onward, I can see and work through the steps. If I were to add these two equations, then I get y = mx + m + 2b - 36. I figure there's substitution occurring for their simplification, but I'm oblivious to it.

edit: Oh jeez, and I posted this in the wrong subsection. Should I re-post it, or will this be moved? Sorry about that!

Hello Lilt. Welcome to PF !

(I have requested that this thread be moved to precalculus.)

If m+b = 36, then b = 36 - m.

Plug-in 36 - m in place of b in y = mx + b.

If you solve for m, you will find a point through which all these lines pass; (lines with equation y = mx+b having m + b = 36).

More pertinent to the question at hand:

The intercept-intercept form of a line is $\displaystyle \frac{x}{a}+\frac{y}{b}=1\,,$ where a is the x-intercept and b is the y-intercept. If you put the equation for your line into this form it may be easier to get the answer.​

 

[Lilt]

I'm still stumbling on the algebra, I think.

So if they solved for b and got b = 36 - m

And plugged that into get y = mx + (36 - m)

They would re-order the equation y = mx - m +36

And then factored out m, y = m(x-1) + 36

But they have y = (m-1)x + 36

Maybe they made an error? And thank you for the welcome :)

 

[SammyS]

I'm still stumbling on the algebra, I think.

So if they solved for b and got b = 36 - m

And plugged that into get y = mx + (36 - m)

They would re-order the equation y = mx - m +36

And then factored out m, y = m(x-1) + 36

But they have y = (m-1)x + 36

Maybe they made an error? And thank you for the welcome :)

B is the y-intercept. Right?

What can you say about m, if the y-intercept has to be an integer?

How do you find the x-intercept from the equation, y = m(x-1) + 36 ?

 

[HallsofIvy]

I don't see any reason to look at y= m(x- 1)+ 36. Since the question asks about the line y= mx+ b= mx+ 36- m, look at its intercepts- when x= 0, y= 36- m which must be an integer. When y= 0, x= (m- 36)/m= 1- 36/m which must be an integer. So m must be an integer that evenly divides 36.

 

[SammyS]

I don't see any reason to look at y= m(x- 1)+ 36. Since the question asks about the line y= mx+ b= mx+ 36- m, look at its intercepts- when x= 0, y= 36- m which must be an integer. When y= 0, x= (m- 36)/m= 1- 36/m which must be an integer. So m must be an integer that evenly divides 36.

Well, if y = 0, then it seems to me that the equation, y = m(x- 1)+ 36 → 0 = m(x- 1)+ 36 lends itself to solving for x, which gives the x intercept. So, if you're looking for a reason, that's one reason.

Another reason to write the equation of the line as y= m(x- 1)+ 36 is: this form of the line gives a family of lines passing through the point, (1, 36), with their slopes parametrized by m.

It's not necessary to use this form for the equation of the line. It's just that in my view it might be helpful.