How would I find parametric representations for the plane:
$2x+3y+z=4$ for $0 \leq x + y + z \leq 7$ and $2 \leq x-y \leq 4$?
I can do simple ones where only $x,y$ are restricted independently (forms a rectangle in the $x-y$ plane, but how would I go about doing this one (when all three variables are involved)?
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Twenty-six colours
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Let $x-y=u$ and $x+y+z=t$. Then you can find $x$, $y$ and $z$ in terms of $u$ and $t$ by Crammer's Rule.
$$(x,y,z)=\left(\frac{2u-t+4}{3},\frac{-u-t+4}{3},\frac{-u+5t-8}{3}\right),$$
where $2\le u\le 4$ and $0\le v\le7$.
CY Aries
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I'm not very familar with Cramer's rule. Is this do-able with just row reduction? – Twenty-six colours May 08 '17 at 11:43
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$4-t=(2x+3y+z)-(x+y+z)=x+2y$. So $(x+2y)-(x-y)=4-t-u$ and hence $3y=4-t-u$. With $y$, you can find $x$ and then $z$. – CY Aries May 08 '17 at 11:47
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Thanks. This is for linear systems. What if I had the condition that $x \geq 0$ and $x^2 + y^2 \leq 4$ which is non-linear? – Twenty-six colours May 08 '17 at 11:49
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Maybe you can let $x=t$, $z=u$ and express $y$ in terms of $t$ and $u$. Rewrite the condition $x^2+y^2\le 4$ in terms of $u$ and $v$. – CY Aries May 08 '17 at 11:52
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So something like $x^2 + y^2$ = $(x+y)^2 - 2xy$? – Twenty-six colours May 08 '17 at 11:54