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How should I explain a $12$ year old that for solving a system of $n$ unknowns we need $n$ equations . I tried to show a example on a graph that for two variables we need two lines that intersect at a point ( it might be the case that they are parallel but we are considering "good" cases ) .

How should I explain this in a simple way ?

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it might be important to know why he/she needs to know that. I imagine there is a particular case, where is is important. Maybe physics? Have you maybe tried the approach that each equation can eliminate one variable? – user667804 Jan 24 '14 at 12:40
You should (probably) add that you're talking about linear equations. For example $$x^2+y^2=0 \qquad x,y\in\mathbb R$$ As one (and only one) solution despite having $2$ unknowns and only one equation. – AlexR Jan 24 '14 at 12:45
up vote 2 down vote accepted

The easiest way I see is to simply explain what a layman that hadn't been taught about row reduction might do. Take your first equation with $n$ unknowns, rearrange it to solve for one of these unknowns and 'plug in' the expression for that unknown into the other equations. Every time you do this, you decrease the amount of unknowns still involved in the existing pool of equations by $1$.

So what happens if there are, say, $n-1$ equations? Well, solving a set of equations (for a unique solution) is essentially just reducing the set of unknowns down to zero. Here you can't reduce the amount of unknowns enough times, so in your last equation you have $ a \ x_{n-1} +b \ x_{n} + c = 0 $ instead of $b \ x_{n} + c = 0 $. But by introducing one more equation you can reduce that line to a single point, providing values for every unknown you previously eliminated from the pool.

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