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Given $(f+g)(x)=10-3x$ and $(f-g)(x)=5x-14$, find $f(x)$ and $g(x)$.

I have no idea where to start, can someone please help me?

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    What did you try? – 2017-01-13
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    You realize that by definition, $(f+g)(x)$ means $f(x)+g(x)$, right? – 2017-01-13

2 Answers 2

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By definition,$$\begin{align*} & (f+g)(x)=f(x)+g(x)\tag1\\ & (f-g)(x)=f(x)-g(x)\tag2\end{align*}$$ And since the problem states $(f+g)(x)=10-3x$ and $(f-g)(x)=5x-14$, we have$$\begin{align*} & f(x)+g(x)=-3x+10\tag3\\ & f(x)-g(x)=5x-14\tag4\end{align*}$$ Which can be easily solved by adding $(3)$ and $(4)$ together to get$$\begin{align*} & 2f(x)=5x-14-3x+10\tag5\\ & \therefore\boxed{f(x)=x-2}\tag6\end{align*}$$ And substituting $(6)$ back into $(3)$ gives$$\begin{align*} & \color{green}{f(x)}+g(x)=-3x+10\tag7\\ & \color{green}{x-2}+g(x)=-3x+10\tag8\\ & \boxed{g(x)=-4x+12}\tag9\end{align*}$$

  • 0
    How is it difficult it's a linear system in $f$ and $g$? Just treat $-3x+10$ as "constant" and $5x-14$ as "constant" when solving. – 2017-01-14
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    @AhmedS.Attaalla Note the possible sarcasm! – 2017-01-14
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    I could never tell...I don't know if sarcasm would be appropriate, we want the op to know exactly what we mean. @Frank – 2017-01-14
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    @AhmedS.Attaalla See my new edit... – 2017-01-14
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    Yes, sarcasm should be avoided to avoid confusion. – 2017-01-14
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    @SimpleArt I removed the sarcasm... – 2017-01-14
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HINT:

What happens if you sum those two results together? Or what if you subtract them?