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Space/Plane equation/Example intersection/Example

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We consider the sets

(from example) and

and we are interested in the intersection

A point lies in this intersection if and only if it fulfills both conditions, that is, both equations (let us call them and ). Does there exist a "simpler“ description of this intersection set? A point that fulfills both equations does also fulfill the equation that arises when we add the equations together, or when we multiply the equation with a number . Such a linear combination of the equations is, for example,

Therefore,

since we can reconstruct the original second equation from the new second equation and vice versa. Hence, the conditions are equivalent. The advantage of the second description is that the variable does not occur in the new second equation; it has been eliminated. Therefore, we can resolve with respect to , and we obtain

For , we must have

Also, these two resolved equations are together equivalent with the original equations, and, therefore, we have

This description yields a more explicit overview of the set .