The determinant decides which of three answers you get. For x + y = 5 and x − y = 1 it is −2, so the lines cross exactly once, at (3, 2). A determinant of zero means the slopes match, and then the constants decide whether the lines are parallel or the same line written twice.
Three outcomes
Two straight lines in a plane can do exactly three things, and a page that reports only the first is answering the easy third of the question.
Cross once. Different slopes. One solution, and it is the common case.
Parallel and distinct. Same slope, different intercepts. They never meet, and the system has no solution.
The same line. Same slope, same intercept, written two different ways. Every point on it is a solution, so there are infinitely many.
In simultaneous-equation language those are the unique, inconsistent and dependent cases. The geometry and the algebra are describing the same three situations, which is worth noticing because the algebra makes it look like arithmetic and the geometry makes it obvious.
What the determinant tests
a₁b₂ − a₂b₁ is the cross product of the two coefficient pairs, and it is zero exactly when those pairs are proportional.
Proportional coefficients mean the same slope, because the slope of ax + by = c is −a/b and scaling both leaves that unchanged.
So the determinant is a parallelism test, dressed as arithmetic. When it is non-zero, Cramer’s rule divides by it — and the closer to zero it is, the larger the coordinates, which is exactly what nearly-parallel lines look like.
Computing it exactly matters more than it might seem. In floating point a determinant that ought to be zero often arrives as a tiny non-zero number, and dividing by that gives an intersection hundreds of billions of units away, reported with complete confidence. On this page the coefficients are kept as fractions and the test is exact.
Why standard form
Both lines are entered as ax + by = c rather than y = mx + c, and that is deliberate.
Slope-intercept form cannot write a vertical line. There is no m for x = 3. Any page built on y = mx + c has to special-case it, and special cases are where the bugs live.
Standard form has no such gap: x = 3 is 1x + 0y = 3, an ordinary set of coefficients requiring no separate branch.
It also makes the determinant natural, since the coefficients are already laid out as a matrix. And the pair (a, b) is the normal vector to the line, which is why perpendicularity can be tested as a₁a₂ + b₁b₂ = 0 — a dot product of zero — and why that test keeps working when one line is vertical.
The angle between them
When the lines cross, they do so at some angle, and there are two of them — supplementary, adding to 180°. The acute one is what gets quoted.
It comes from the difference of the slopes: tanθ = |(m₂ − m₁)/(1 + m₁m₂)|.
The denominator is the interesting part. When it is zero, the tangent is undefined and the angle is 90° — which is the m₁m₂ = −1 perpendicularity condition arriving from a different direction.
This page does not use that formula for the perpendicular test, because it needs both slopes to exist. The dot product of the normals is zero in exactly the same cases and keeps working when one line is vertical.
Where it gets used
Simultaneous equations. Every two-variable system is this problem. The three outcomes are the three things a system can do.
Linear programming. The corners of a feasible region are intersections of constraint lines, and the optimum sits at one of them.
Computer graphics and games. Ray casting, collision detection and clipping are all line intersections, run millions of times a second — which is why the degenerate cases have to be handled rather than hoped past.
Surveying and navigation. Fixing a position from two bearings is finding where two lines cross.
Sources and methodology
Cramer’s rule and the degenerate cases are standard; these are the references.
Method. The determinant is computed on exact rationals, which is not a precision nicety: its being zero is the whole classification, and a value that should be exactly zero arriving as 1e−17 would send the page down the crossing branch and produce coordinates the size of a galaxy. Perpendicularity is tested as a₁a₂ + b₁b₂ = 0 — the dot product of the two normals — rather than through a product of slopes, because that product does not exist when a line is vertical. The suite substitutes every returned intersection back into both original equations. That engine is verified on every change against 119 hand-written assertions, including that every reported intersection satisfies both original equations by substitution across every non-degenerate integer coefficient pair in a seven-by-seven grid. The count and the per-case breakdown are published on the formula verification page.
Read the guide
With three or more unknowns the same determinant idea generalises — the System of Equations Calculator handles the larger cases.