We can solve for 2 unknowns from one equation: Is it possible to solve for two unknowns from one equation?
But is it possible to solve for n unknowns from one equation?
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2Yes – sometimes – if the variables are required to be integers. – Gerry Myerson Jun 24 '21 at 10:27
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What do you mean by "solve"? – Jun 24 '21 at 10:28
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@Gae.S. Find all unknowns. – sbh Jun 24 '21 at 10:29
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Why does this apply to the equation you've linked? I would say it does not. – Jun 24 '21 at 10:29
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In general, no and most single equations you encounter such as the linked $x+3y=32$ have multiple solutions. But there are special cases.
If you are restricted to real numbers, then it is possible to transform simultaneous equations into a single equation.
For example $$x+y+z=6$$ $$x+2y+4z=17$$ $$3x+y+2z=11$$ have the unique solution $x=1,y=2,z=3$. So too does the single equation $$(x+y+z-6)^2+(x+2y+4z-17)^2+(3x+y+2z-11)^2=0$$ which you could write as the less obvious $$21z^2+6y^2+11x^2+22yz+22xz+12xy-192z-102y-112x+446=0.$$
In a sense this is cheating, and there are an infinite number of solutions involving complex numbers.
Henry
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