A system of equations consisting of a linear equation and a quadratic equation has infinitely many solutions. Is the statement true? Can the statement be true even sometimes?
2026-04-12 03:53:29.1775966009
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Does a system of equations consisting of a linear equation and a quadratic equation have infinitely many solutions?
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If you work in the projective plane $\mathbf P^2(\mathbf C)$, the intersection of the corresponding algebraic varieties is in general a finite set, and consists of $2$ points, unless the line is a component of the conic.
In greater dimensions, say in $\mathbf P^r(\mathbf C)$, you get an algebraic variety of dimension $\ge r-2$, hence it has an infinite number of points.
No, not at all. Consider a line and a circle. The intersection can atmost be 2 points. Or a line and a parabola for that matter.