# 2015 STEP I Question 2 (Assignment 16)

I'm fine for the first two parts of the question but for the last part, I'm just wondering how it was found in the solutions that y=2x was a suitable substitution to make so that the equation would work. Is there a method or should you be able to know by just looking at it or by trial and error? Thanks.

### Substitution

It's a bit trial and error-ish.

One of the approaches you can make it to start by noting that the cubics have the same structure (no squared terms etc). This means that a substitution of the form $y=kx$ (rather than $y=kx+c$) might be suitable.

If you substitute $y=kx$ into the second equation you get:
$y^3-3y-\sqrt{2}=0\\ k^3x^3 - 3kx - \sqrt{2} =0\$
Then we want to divide by $k$ so that the coefficient of $x$ is correct which gives:
$k^2x^3-3x - \tfrac 1 k \sqrt{2} = 0$

Finally, comparing this to the first equation in $x$ suggests that $k=2$ is a sensible thing to try.

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