Relating a System's Solutions to Its Graphs
A system's solutions are exactly the points where its graphs intersect, so counting crossings counts solutions, and the discriminant tells you that count without graphing.
A solution to a system of two equations is a point that satisfies both equations at once. On a graph, that means the point sits on both curves. So wherever the two graphs cross or touch, that point is a solution, and wherever they don't meet, there is no solution there. This is why the number of intersection points equals the number of solutions: the graphs are just pictures of every (x, y) pair that makes each equation true, and a shared point is a pair that makes both true.
Picture a line sliding straight down across a parabola that opens upward. Start high above the parabola: the line misses it completely, zero crossings, zero solutions. Lower the line and it becomes tangent, grazing the parabola at exactly one point: one solution. Lower it further and it slices through, cutting the parabola at two points: two solutions. Algebraically, substituting one equation into the other collapses the system into a single quadratic, and its discriminant b2 - 4ac tracks this same story: negative means no real crossing, zero means exactly one (tangency), positive means two.
This connects directly to solving systems by substitution: you turn two equations into one, then read off how many real solutions it has before even finding them. The same idea extends past parabolas. Swap the parabola for a circle and a line can miss it, graze it tangentially, or pierce it at two points, same three outcomes, same discriminant logic, just applied to the equation substitution produces.
Key terms
- System of equations
- A set of equations considered together, whose solutions are the points satisfying all of them at once.
- Discriminant
- The quantity b2 - 4ac from a quadratic, whose sign tells how many real solutions the quadratic has.
- Tangent
- Describes a line and curve that touch at exactly one point without crossing through.
- Substitution
- Replacing one variable with an equivalent expression from another equation to combine a system into one equation.
Watch it explained
- How it worksHow does a line sliding down across a parabola go from zero crossings to one to two?
- Worked exampleFind how many real solutions the system y = x2+2 and y = x-1 has, using the discriminant.
- Common mistakeWhy isn't a spot where two graphs come close but never touch a solution of the system?
- Going furtherWhat changes about the picture when the nonlinear equation is a circle instead of a parabola?
Check yourself
End of topic.