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.
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1$\begingroup$ The new variable here is $y$. I believe it's asking you to replace all $x$ with the new variable $y$ based on the relationship $y=x-2$. In other words, rewrite $x^2-3x+5=0$ except using $y$ instead of $x$. $\endgroup$– Serendipitous EpiphanyFeb 11, 2020 at 8:10
2 Answers
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
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 $y=x-2$.
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.