How to solve a system of two equations in three unknowns

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$x+y=5$

$2x+y-3z=12$

I know that in order to solve three unknowns three equations are needed, so I'm unsure if this can be solved or if different techniques (apart from the usual eliminations/substitutions) are used to solve problems like these.

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3 Answers

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This system of equations can be solved, it just doesn't have a unique solution.

We can try to solve it as if we had just 2 variables, $x$ and $y$.

Subtract the first equation from the second to get $x = 3z + 7.$

Then since $y = 5 - x$ we get $y = -3z - 2.$

We've now solved the system, we just have one solution for each possible choice of $z$.

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A solution is to move one of the unknowns to the RHS and consider the system of equations as a parametric one.

$$\begin{align}x+y&=5\\ 2x+y&=12+3z.\end{align}$$

You now solve the $2\times2$ system, giving

$$\begin{align}x&=7+3z\\ y&=-2-3z.\end{align}$$

There is no restriction on the values that $z$ can take, giving a complete set of solutions.

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Subtracting one from the other you are left with a relation between two variables e.g.,

$$ x - 3 z = 7 $$

Without one more (x,z) relation there's no way you can proceed towards a solution.

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