Find the point where two lines, each given in slope-intercept form (y = mx + b), cross.
How it works
Setting the two equations equal and solving gives x = (b₂ – b₁) ÷ (m₁ – m₂), then substituting back finds y. The lines y = 2x + 1 and y = -x + 10 cross at (3, 7).
What this does not include
This handles lines given in slope-intercept form. Lines given in a different form (like standard form Ax + By = C) would need converting to slope-intercept form first.
How to use this calculator
- Enter the slope and intercept of the first line.
- Enter the slope and intercept of the second line.
A worked example
Line 1: y = 2x + 1. Line 2: y = −x + 10. Intersection: (3, 7).
Line 1: y = x. Line 2: y = 3x + 4. Intersection: (−2, −2).
What the variables mean
| Variable | Meaning |
|---|---|
| m1, b1 | Slope and intercept of the first line |
| m2, b2 | Slope and intercept of the second line |
Edge cases worth knowing
Parallel lines (equal slopes) never intersect — unless they’re the exact same line, there’s no single (x, y) point that satisfies both equations, so the calculator declines to show a result.
This solves the system algebraically, not graphically — setting both equations equal and solving for x, then substituting back to find y, rather than estimating from a plotted graph.
What does it mean if two lines have the same slope?
They’re parallel and never cross (unless they’re actually the identical line, in which case every point is a shared point) — this calculator returns no single-point result in either case.
Can two lines intersect at more than one point?
No — two distinct straight lines intersect at exactly one point, unless they’re parallel (no intersection) or identical (infinitely many shared points).
How is this used in practical problems?
Finding where two linear relationships meet is common in break-even analysis, supply-and-demand graphs, and any problem comparing two straight-line trends.
