What does it mean

Question

Rewrite the following equation in terms of the new variable $$ x^2-3x+5=0 ,\qquad y=x-2 $$ The lines don't even meet(one is straight another one is curve line) I don't understand what new variable means.

Answer 1

It is asking you to rewrite the equation to be in terms of $y$ instead of in terms of $x$.

To do this, we first need to find out what $x$ is in terms of $y$, that is $x=y+2$

Then we put $y+2$ into the original equation everywhere that we see an $x$

\begin{align}x^2-3x+5&=0\\ (y+2)^2-3(y+2)+5&=0\end{align}

Finally, we need so simplify this new equation by expanding the brackets and collecting like-terms. I will let you do this part

Answer 2

The new variable is just that. A variable, that is, a symbol representing a number, which was not previously seen. "Previously" meaning in all the expressions so far.

In particular, what you need to do is to find an expression that 1. depends entirely on $y$, and not on $x$, 2. Is true if and only if the original expression is true.

In other words, you are given the equation $f(x)=0$, and you need to find some function $g(y)$ such that $g(y)=0\iff f(x)=0$, where the connection between $x$ and $y$ is $y=x-2$.

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In order to do that, you can go one of two ways.

The simpler way would be to express $x$ as a function of $y$, and replace all instances of $x$ with that function.

Alternatively, you can guess that the new expression will, just like the old, be a quadratic polynomial, and so will equal $ay^2+by+c$. Now, replace $y=x-2$, and compare the relevant coefficients.