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I was wondering how you would write the function $y=\frac{4x-7}{x+1}$ in the form of $\frac{a}{x+1}+k$.

I know the vertical asymptote is at $x=-1$.

I know that to find a (the dillation) that you usually need a point. The asymptotes are usually given in the question, but not this time. I also know how to change the form with linear and quadratic functions, but I seem to be getting stuck with these types of questions.

I tried solving algebraically, but I seem to always get stuck. I'm not sure what I should do first. I've tried plotting points but I cannot figure out what the equation would be from the graph.

When I mean algebraically, I mean that I tried bringing $x+1$ over to the other side, so that it would become $y = \frac{4x-7}{x+1}+k.$ However I do not know where I should go from here.

Any help would be appreciated. Thank you very much.

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    $\begingroup$ No trick is needed. $$\frac{a}{x+1}+k=\frac{k(x+1)+a}{x+1}$$ so you just have to solve $$k=4,\quad k+a=-7.$$ $\endgroup$ Commented May 31, 2023 at 6:08
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    $\begingroup$ "I tried solving algebraically, but I seem to always get stuck": Please edit your post to include your attempts. Quick beginner guide for asking a well-received question + please avoid "no clue" questions. $\endgroup$ Commented May 31, 2023 at 6:12
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    $\begingroup$ @AnneBauval edited, thank you. $\endgroup$ Commented May 31, 2023 at 6:23
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    $\begingroup$ Your edit ($y - k= \frac{4x-7}{x+1}$) is not consistent with the context ($y=\frac{4x-7}{x+1}=\frac{a}{x+1}+k$). Did you follow my first comment, instead? $\endgroup$ Commented May 31, 2023 at 7:13

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$$y=\frac{4x-7}{x+1}=\frac{4x+4-11}{x+1}$$ $$\implies y= \frac{4(x+1)-11}{x+1}$$ $$\implies y=4-\frac{11}{x+1}$$ Here $a=-11$ and $k=4$

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  • $\begingroup$ @verygood101 I didn't get that, i got $$4x-7=4x+4-11$$ as we know that $7=11-4$ so $-7=4-11$ $\endgroup$ Commented May 31, 2023 at 6:27

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