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A Question on an Interesting Method to Solve a System of Equations

 
 
Picture of Tina Jin
A Question on an Interesting Method to Solve a System of Equations
by Tina Jin - Monday, 15 January 2024, 9:40 AM
 

Hello,

For systems like 3.34 on MCIII Algebra with no constant term:

x^3+y^3+20xy=0

x^2+y^2+18x=0

I was presented with a method that set y=kx, but how do you know y is not kx+n for a constant n?

I know there's no constant term, but if multiplied by x then kx^2+nx, which means there's no constant term anymore.

More generally, does this method of setting a variable as k times the other variable always work for equations with no constant term?

Thanks,

Tina Jin

 
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Re: A Question on an Interesting Method to Solve a System of Equations
by Dr. Kevin Wang - Monday, 15 January 2024, 11:05 PM
 

This method is in fact a change of variable, where we set a new variable $k = y/x$ and convert the problem to an equation for $k$.  Remember that the solution $x$ and $y$ are just numbers, and the variable $k$ simply represents their ratio (as long as $y$ is not $0$).  For multiple solutions for $(x,y)$, we have multiple solutions for $k$.  This is not about $y$ and $x$ (as variables) having a linear relation or something like that.  It is simply a method to convert the problem to solving for a new unknown variable.

If there is a constant term on the right hand side, the change of variable $y=kx$ is still valid, but the resulting new equation is usually not easy to simplify.  Here we don't have a constant term, and we can divide by $x$ and get two simpler equations.  This idea could also work (to simplify the equations) in the general case you mention.

Picture of Tina Jin
Re: A Question on an Interesting Method to Solve a System of Equations
by Tina Jin - Tuesday, 16 January 2024, 8:58 PM
 

Oh, I forgot that x and y are just numbers, and two numbers have a ratio.

Thanks!