Interactive guide
A "solution" is just where two lines cross.
Drag any of the four points. Each line follows the two points that define it.
equation 1 equation 2 solution
2.00x + 1.00y = 1.00 2.00x − 1.00y = −3.00
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- Solution
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- det
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Each line is every (x, y) that satisfies one equation. A point that satisfies both equations has to sit on both lines at once, which means it has to be exactly where they cross. Drag any of the four points below, each line is defined by the two points on it, and watch that crossing point slide, split apart, or disappear entirely.
Three outcomes, no others
Two lines in a plane can only relate to each other three ways: they cross exactly once, they never cross, or they're the same line. There's no fourth option, and which of the three you get is decided entirely by one number: the determinant of the equations' coefficients.
det = a1b2 − a2b1, built only from how steep each line is, never from where it sits. That's exactly why parallel lines (same steepness, different position) always land on det = 0, whether or not they happen to be the same line.
Reading the picture
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1
det ≠ 0: exactly one crossing
Different slopes always meet somewhere. Solve with Cramer's rule, straight from the two equations.
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2
det = 0, lines distinct: no solution
Same slope, different intercept. They run side by side forever and never touch.
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3
det = 0, lines identical: infinite solutions
Every point on the line satisfies both equations, they were never two constraints, just one written twice.
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4
det = 0 either way looks identical numerically
You have to check the intercepts separately to know which of the two you actually have.
That determinant, a1b2 − a2b1, is exactly the same test from the Matrix Transformations page. Stack the two equations' coefficients into a matrix, [[a1, b1], [a2, b2]], and this whole system is just Mv = c: a singular M there is a no-solution-or-infinite-solutions system here, same number, same reason, just a different question asked of it.
The jargons
Two lines, one number that decides how they relate.
A set of equations considered together; a solution has to satisfy every one of them at once.
One with at least one solution, whether that's exactly one point or an entire shared line.
One with no solution at all, the geometric picture of two parallel lines that never meet.
The a's and b's from both equations, arranged as a matrix; its determinant decides which of the three outcomes you're in.
A direct formula for x and y as ratios of determinants, no elimination or substitution needed, as long as det ≠ 0.
Equations that describe the same line. One is redundant, a scaled copy of the other in disguise.