Simultaneously Solve a Linear and Cubic equation

[bartrocs]

Homework Statement

Solve the following equations Simultaneously:
y=x^3 and y=3x-2

My question is, how do I solve this? I've tried everything

The Attempt at a Solution

I've tried to isolate x in the linear equation and substitute it into the cubic. I don't know if that's wrong but it didn't work for me. This is what I did:

isolate x in linear equation: x= -2/3-y/3
substitute this into cubic equation: y=(-2/3-y/3)^3
I then expanded this and it didn't work.

I also tried: 3x-2=x^3
therefore: x^3-3x+2=0
this didn't work either! please help!

 

[Bohrok]

I also tried: 3x-2=x^3
therefore: x^3-3x+2=0
this didn't work either! please help!

This way is a good start; do you know about the rational root theorem?

 

[SteamKing]

By inspection, there is at least one integer solution between 0 and 2 to the two equations.
There is also another integer solution between -3 and -1.

 

[bartrocs]

By inspection, there is at least one integer solution between 0 and 2 to the two equations.
There is also another integer solution between -3 and -1.

I would prefer you told me the methods for solving the equations rather than the answer. I already know the answer by using my graphics calculator. However this is of no use to me as I want to solve it the proper way without a calculator.

do you know about the rational root theorem?

I didn't. But I Googled it and I do know now. Is there any way to do these simultaneously like you can with a quadratic and linear equation?

 

[ehild]

Try to group the terms so as something can be factor out.

x³-3x-2=(x³-x)-2(x-1).

The rational root theorem is also very useful. It states: For a polynomial equation
a_nxⁿ+a_n-1x^n-1+...a₀=0,

if p is an integer factor of the constant term a₀, and q is an integer factor of the leading coefficient a_n, the rational solution is of the form ±p/q.

In this case, a₃=1 and a₀=-2, so q=1, and the possible p values are 1 and 2. So you have 4 possible values for x to try if they are roots or not. If one of them (x1) is a root, you can divide the equation by the factor of (x-x1) and you get a quadratic equation to solve.


ehild

 

[SteveL27]

I would prefer you told me the methods for solving the equations rather than the answer. I already know the answer by using my graphics calculator. However this is of no use to me as I want to solve it the proper way without a calculator.

Believe it or not, inspection is actually the first thing you should always try when given a problem like this. Inspection is an entirely proper technique.

There is some meta-reasoning involved. They can't actually want to make you solve a random cubic, since that would be too difficult. There might be a simple solution. So you mentally plug in the simplest numbers ... 0, 1, -1, 2, etc. And if you do this, you find a solution.

Once you have one solution, you can do a polynomial division to reduce the cubic to a quadratic, which you know how to solve using the quadratic formula.

So even though there are a few more technical things you can and should try; the very first thing you should do is plug in the most obvious simple numbers and see if a solution pops out.

 

[PeterO]

I also tried: 3x-2=x^3
therefore: x^3-3x+2=0
this didn't work either! please help!

That last expression can be easily factorised into 3 linear factors, and I am sure that was supposed to be the next step in the solution.

 

[NascentOxygen]

therefore: x^3-3x+2=0

I can see that x=1 will make the LHS = 0, so x=1 is a solution.

Now, take out the factor (x-1).

In your working for this, you divide x³-3x+2 by (x-1)