How this instrument works
Two non-vertical lines, y = m₁x + b₁ and y = m₂x + b₂, share a single point wherever both equations give the same y at the same x. Setting them equal, m₁x + b₁ = m₂x + b₂, collects into x(m₁ − m₂) = b₂ − b₁, which rearranges to x = (b₂ − b₁) ⁄ (m₁ − m₂). Substituting that x into either original line then recovers y. The name 'corner point' comes from linear programming, where a feasible region is bounded by several such lines and each vertex of that region is exactly one of these pairwise crossings.
The shape of the formula follows from a single move: subtracting one line's equation from the other cancels y entirely, leaving one equation in one unknown, x. That is the substitution method for a 2×2 linear system, done by hand rather than with a matrix. It is also the arithmetic underneath the fundamental theorem of linear programming — if a linear objective has an optimum over a convex feasible region, that optimum always sits at one of the region's corners, never floating in the open interior, so a solver only ever needs to test the vertices, not the whole shaded area.
One case breaks the formula cleanly: when m₁ = m₂, the denominator m₁ − m₂ is zero and no finite crossing exists — the lines run parallel. If the intercepts also match, the two entries describe the identical line, and every point on it satisfies both equations at once rather than none. A second, subtler limit worth knowing: an algebraic crossing of two boundary lines is only a candidate corner. In a region built from more than two constraints, that point must still satisfy every other inequality to count as an actual vertex.
- Enter the first boundary line's numbers into Line 1 slope, m₁ and Line 1 intercept, b₁.
- Enter the second line's numbers into Line 2 slope, m₂ and Line 2 intercept, b₂, using the same units as the first line.
- Read Intersection: x for the horizontal coordinate where the two lines cross.
- Read Intersection: y — found by substituting that x back into Line 1's equation.
- If m₁ equals m₂, the lines are parallel and never meet; the sheet flags this instead of dividing by zero.
Worked example — two lines meeting at (2, 5)
Take the lines y = 2x + 1 and y = −x + 7, entered as m₁ = 2, b₁ = 1, m₂ = −1, b₂ = 7. Setting the two expressions equal gives 2x + 1 = −x + 7, so 3x = 6 and x = (7 − 1) ⁄ (2 − (−1)) = 6 ⁄ 3 = 2 exactly, matching the sheet's Intersection: x reading of 2.
Substitute x = 2 back into either line to find the height: y = 2(2) + 1 = 5, and the second line agrees, y = −1(2) + 7 = 5 — a two-line check that catches a mistyped slope before it produces a wrong vertex. Picture a feasible region bounded above by y ≤ 2x + 1 and y ≤ −x + 7: the crossing at (2, 5) is exactly the corner an optimizer would test, since the best value of any linear objective over that region sits at one of its corners, never in the open interior.
Questions
Why is this called a corner point rather than just an intersection?
Corner point is the name linear programming gives to a vertex of a feasible region, the shape bounded by several straight constraint lines. Each vertex is where two of those boundary lines cross, exactly the calculation this sheet performs. The fundamental theorem of linear programming says that if an optimal solution exists, it always sits at one of these corners, never inside the open region.
How is the intersection formula derived?
Set the two line equations equal, since at the crossing point both give the same y for the same x: m₁x + b₁ = m₂x + b₂. Collecting the x terms gives x(m₁ − m₂) = b₂ − b₁, so x = (b₂ − b₁) ⁄ (m₁ − m₂). Substituting that x back into either original line, y = m₁x + b₁, recovers the matching height, and both lines must agree on it.
What happens if the two slopes are equal?
If m₁ = m₂ the lines are parallel: the denominator m₁ − m₂ is zero and no finite crossing exists, so the sheet raises a flag instead of dividing by zero. If the intercepts also match, b₁ = b₂, the two entries describe the same line, and every point on it is a solution rather than none.
Does every intersection found this way sit on a feasible region's boundary?
Not automatically. This sheet finds where two lines cross algebraically, which is the first step, but a feasible region usually has more than two boundary constraints. A genuine corner point must also satisfy every other inequality in the system; a crossing that fails even one of them is a candidate that gets discarded, not a true vertex.
Why check the answer against both lines instead of just one?
Because it costs nothing and catches mistakes. y = m₁x + b₁ and y = m₂x + b₂ must produce the identical number at a true intersection, so computing both and comparing them flags a transposed digit or a sign slip immediately — if the two disagree, one of the four inputs is wrong, not the arithmetic that follows it.
Who actually uses corner points in practice?
Anyone solving a linear program by hand or checking a solver's output. The simplex method, developed by George Dantzig in the 1940s, walks from corner to corner of a feasible region rather than searching its interior, because the optimum is guaranteed to live on one of them. The two-line intersection worked here is the arithmetic underneath every step of that walk.