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Question Number 129979 by liberty last updated on 21/Jan/21

$$\mathrm{If}\{\begin{array}{l}\mid \mathrm{x}\mid +\mathrm{x}+\mathrm{y}=5\\ \mathrm{x}+\mid \mathrm{y}\mid -\mathrm{y}=10\end{array}\mathrm{then}\mathrm{x}+\mathrm{y}?$$

Answered by MJS_new last updated on 21/Jan/21

$$\mid x\mid +x+y=5$$$$x⩽0\Rightarrow y=5$$$$\mathrm{but}\mathrm{then}\mathrm{the}{2}^{\mathrm{nd}}\mathrm{eq}.\mathrm{gives}x=10>0$$$$\Rightarrow$$$$x>0$$$$$$$$x+\mid y\mid -y=10$$$$y⩾0\Rightarrow x=10$$$$\mathrm{but}\mathrm{then}\mathrm{the}{1}^{\mathrm{st}}\mathrm{eq}.\mathrm{gives}y=-15<0$$$$$$$$\Rightarrow x>0\wedge y<0$$$$\mathrm{now}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{easy}\mathrm{to}\mathrm{solve}$$

Answered by EDWIN88 last updated on 21/Jan/21

$$\mathrm{If}\mathrm{y}⩾0\mathrm{then}\mathrm{the}\mathrm{second}\mathrm{eq}\mathrm{tells}\mathrm{us}\mathrm{that}$$$$\mathrm{x}=10,\mathrm{but}\mathrm{substituting}\mathrm{this}\mathrm{in}\mathrm{the}\mathrm{first}\mathrm{eq}$$$$\mathrm{we}\mathrm{find}\mathrm{that}\mathrm{y}=-15\mathrm{contracting}\mathrm{that}\mathrm{y}⩾0$$$$\mathrm{we}\mathrm{conclude}\mathrm{that}\mathrm{y}<0.\mathrm{Then}\mathrm{it}\mathrm{cannot}$$$$\mathrm{be}\mathrm{that}\mathrm{x}⩽0\mathrm{because}\mathrm{that}\mathrm{leads}\mathrm{to}\mathrm{y}=5$$$$\mathrm{we}\mathrm{conclude}\mathrm{that}\mathrm{x}>0$$$$\mathrm{Then}\mathrm{we}\mathrm{find}\mathrm{the}\mathrm{following}\mathrm{eq}\{\begin{array}{l}2\mathrm{x}+\mathrm{y}=5\\ \mathrm{x}-2\mathrm{y}=10\end{array}$$$$\mathrm{we}\mathrm{get}(\mathrm{x},\mathrm{y})=(4,-3)\mathrm{so}\mathrm{x}+\mathrm{y}=1$$

Answered by mathmax by abdo last updated on 21/Jan/21

$$\Rightarrow \{\begin{array}{l}\mathrm{A}(\mathrm{x},\mathrm{y})=5\mathrm{with}\mathrm{A}(\mathrm{x},\mathrm{y})=\mid \mathrm{x}\mid +\mathrm{x}+\mathrm{y}\mathrm{and}\mathrm{B}(\mathrm{x},\mathrm{y})=\mathrm{x}+\mid \mathrm{y}\mid -\mathrm{y}\\ \mathrm{B}(\mathrm{x},\mathrm{y})=10\end{array}$$$$\mathrm{case}1\mathrm{x}⩾0\mathrm{and}\mathrm{y}⩾0\mathrm{s}\Rightarrow \{\begin{array}{l}2\mathrm{x}+\mathrm{y}=5\Rightarrow \{\begin{array}{l}\mathrm{x}=10\\ \mathrm{y}=-15\mathrm{not}\mathrm{solution}\end{array}\\ \mathrm{x}=10\end{array}$$$$\mathrm{case}2\mathrm{x}⩾0\mathrm{y}⩽0\mathrm{s}\Rightarrow \{\begin{array}{l}2\mathrm{x}+\mathrm{y}=5\Rightarrow \{\begin{array}{l}2\mathrm{x}+\mathrm{y}=5\\ 2\mathrm{x}-4\mathrm{y}=20\Rightarrow \{\begin{array}{l}5\mathrm{y}=-15\\ \mathrm{x}=2\mathrm{y}+10\end{array}\end{array}\\ \mathrm{x}-2\mathrm{y}=10\end{array}$$$$\Rightarrow \{\begin{array}{l}\mathrm{y}=-3\\ \mathrm{x}=4\end{array}$$$$\mathrm{case}3\mathrm{x}⩽0\mathrm{and}\mathrm{y}⩾0\mathrm{s}\Rightarrow \{\begin{array}{l}\mathrm{y}=5\\ \mathrm{x}=10\mathrm{not}\mathrm{solution}!\end{array}$$$$\mathrm{case}4\mathrm{x}⩽0\mathrm{and}\mathrm{y}⩽0\mathrm{s}\Rightarrow \{\begin{array}{l}\mathrm{y}=5\\ \mathrm{x}-2\mathrm{y}=10\Rightarrow \{\begin{array}{l}\mathrm{y}=5\\ \mathrm{x}=20\mathrm{not}\mathrm{solution}\end{array}\end{array}$$

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