How can I find an intersect of a function of two variables and a linear function?

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pineapplewithmouse

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I have the function f(x,y)=x^3+xy^2+y^2 and g(x)=-1.84x. I want to find where they are intersecting.
Apparently I need to use something that's called "numerical methods" which seems very difficult and confusing.
I tried to see it on geogebra but for some reason I can't see the exact intersections, but I can see their approximate coordinates, which is not enough.
Is there a method that I can use to get an answer?
The approximate coordinates that I got:
(-0.77, 1.41, 0)
(0.03, -0.06, 0)
(-0.04, 0.08, 0)
 
There is not just one numerical method. Which one have you used in class? To receive help you need to at least tell us the method(s) you can use and show us an attempt at using that method.
 
I didn't learned that.
I wanted to learn how to solve it so I searched on the internet and it said to use "numerical methods".
 
Define h(x,y) = f(x,y)-g(x). Suppose you can see on a graph that a root is between x=1 and x=2.
Now try x=1.5. Is the root between 1 and 1.5 or between 1.5 and 2?
Continue this process until you have nailed the root down to an interval that you feel is close enough to the actual root.
 
If two values are equal, then their difference is 0!

f(x,y) = g(x) exactly where f(x,y) - g(x) = 0. Come on, you need to think.
Now it is your turn to show some work.
 
It weird, h(x,y) equal to 0 just in (0, 0, 0).
So maybe the (0.03, -0.06, 0) (-0.04, 0.08, 0) are just a visual bug of the software, but g(x) clearly goes through f(x,y) in like (-0.77, 1.41, 0).
 

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pineapplewithmouse said:
I have the function f(x,y)=x^3+xy^2+y^2 and g(x)=-1.84x. I want to find where they are intersecting.
Apparently I need to use something that's called "numerical methods" which seems very difficult and confusing.
I tried to see it on geogebra but for some reason I can't see the exact intersections, but I can see their approximate coordinates, which is not enough.
Is there a method that I can use to get an answer?
The approximate coordinates that I got:
(-0.77, 1.41, 0)
(0.03, -0.06, 0)
(-0.04, 0.08, 0)
Click to expand...
I'm not sure exactly what you are trying to do. Functions don't intersect; their graphs do. So I suppose you are talking about the surfaces z=x^3+xy^2+y^2 and z=-1.84x, in three dimensions, which will intersect not in a point, but a curve. Is that what you have in mind?

1661278427159.png​

Of course, if that was what you mean, you would have said g(x,y), not g(x).

Please show us what you did in GeoGebra, and what the three points you show are.
 
After playing a little bit with the graphs, I made a conclusion that the intersection is approximately at (1.42, (-71/92), 0)
 
pineapplewithmouse said:
No, I didn't meant z=-1.84x, I meant y=-1.84x.
Click to expand...
But then, you're trying to solve this system of two equations in three variables:

z=x^3+xy^2+y^2​
y=-1.84x​

All you can do is to substitute in the first equation and solve for z as a function of x:

z=x^3+x(-1.84x)^2+(-1.84x)^2​
​
z=4.3856x^3+3.3856x^2=(4.3856x+3.3856)x^2​

You can't solve for individual points; just set x to whatever value you want, and find what y and z are. Again, the solution is a curve:

1661281613091.png​

A parametric equation of this curve is

x=t​
y=-1.84t​
z=(4.3856t+3.3856)t^2​

If you want z=0, you can solve and find two point where z=0, namely t=0, point (0,0,0) and t=-0.772, point (-0.772,1.42,0). You seem to have swapped some variables, unless I did.

What we really need to get from you is what the original problem is. Why are you trying to do this? That is what will determine what you really want to do.

One more thing: who told you you need numerical methods?
 
Dr.Peterson said:
What we really need to get from you is what the original problem is. Why are you trying to do this? That is what will determine what you really want to do.
Click to expand...
There is a game with a big quest to find some secret items.
The most recent clue that we have is an item these 2 equations.
I am trying to understand what can they mean, so I thought maybe intersecting them will give something.
In the game, the second equation is the vector u=(-125)i+(230)j. It's located on the line -1.84x because I thought it will help.

Dr.Peterson said:
One more thing: who told you you need numerical methods?
Click to expand...
A random person on math.stackexchange.com said that if you have 2 multivariable functions you may want to use numerical methods to solve this system of equations.
 
pineapplewithmouse said:
There is a game with a big quest to find some secret items.
The most recent clue that we have is an item these 2 equations.
I am trying to understand what can they mean, so I thought maybe intersecting them will give something.
In the game, the second equation is the vector u=(-125)i+(230)j. It's located on the line -1.84x because I thought it will help.


A random person on math.stackexchange.com said that if you have 2 multivariable functions you may want to use numerical methods to solve this system of equations.
Click to expand...
Maybe you should just show the actual problem, so we can try to interpret it as written, rather than work back from your misinterpretation? This is a fairly common problem on this site, that people think they're helping us by not giving the real problem.

But your still-vague description suggests that you mean the line y = -1.84x, z=0, rather than the plane y = -1.84x. So my comment about wanting z=0 would be relevant.

pineapplewithmouse said:
How you made it to show the intersect curve of both functions?
Click to expand...
There is an Intersect Two Surfaces tool (under Curves).
 
Dr.Peterson said:
Maybe you should just show the actual problem, so we can try to interpret it as written, rather than work back from your misinterpretation? This is a fairly common problem on this site, that people think they're helping us by not giving the real problem.

But your still-vague description suggests that you mean the line y = -1.84x, z=0, rather than the plane y = -1.84x. So my comment about wanting z=0 would be relevant.
Click to expand...
There is no problem, the only thing I have is:
"f(x,y)=x^3+xy^2+y^2, u=(-125)i+(230)j"
That's it, I need to figure out what I need to do with it.
 
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