{
    "name": "Line Intersection Calculator",
    "url": "https://calculators.geekmidst.com/calculators/line-intersection-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/line-intersection-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Find the point where two lines cross from slope-intercept or ax + by = c form, the angle between them, and whether they are parallel or the same line.",
    "summary": "Two lines in a plane meet at exactly one point unless they are parallel. Solving the two equations together gives that point.",
    "also_called": [
        "Intersection point calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "form",
            "label": "Equation form",
            "type": "segmented",
            "default": "slope",
            "options": {
                "slope": "y = mx + b",
                "standard": "ax + by = c"
            }
        },
        {
            "name": "m1",
            "label": "Line 1: slope m₁",
            "type": "number",
            "default": "2",
            "shown_when": {
                "form": "slope"
            }
        },
        {
            "name": "k1",
            "label": "Line 1: intercept b₁",
            "type": "number",
            "default": "1",
            "shown_when": {
                "form": "slope"
            }
        },
        {
            "name": "m2",
            "label": "Line 2: slope m₂",
            "type": "number",
            "default": "-1",
            "shown_when": {
                "form": "slope"
            }
        },
        {
            "name": "k2",
            "label": "Line 2: intercept b₂",
            "type": "number",
            "default": "4",
            "shown_when": {
                "form": "slope"
            }
        },
        {
            "name": "a1",
            "label": "Line 1: a₁",
            "type": "number",
            "default": "1",
            "shown_when": {
                "form": "standard"
            }
        },
        {
            "name": "b1",
            "label": "Line 1: b₁",
            "type": "number",
            "default": "1",
            "shown_when": {
                "form": "standard"
            }
        },
        {
            "name": "c1",
            "label": "Line 1: c₁",
            "type": "number",
            "default": "5",
            "shown_when": {
                "form": "standard"
            }
        },
        {
            "name": "a2",
            "label": "Line 2: a₂",
            "type": "number",
            "default": "1",
            "shown_when": {
                "form": "standard"
            }
        },
        {
            "name": "b2",
            "label": "Line 2: b₂",
            "type": "number",
            "default": "-1",
            "shown_when": {
                "form": "standard"
            }
        },
        {
            "name": "c2",
            "label": "Line 2: c₂",
            "type": "number",
            "default": "1",
            "shown_when": {
                "form": "standard"
            }
        }
    ],
    "outputs": [
        {
            "key": "point",
            "label": "Intersection point (x, y)",
            "format": "text"
        },
        {
            "key": "status",
            "label": "Result",
            "format": "text"
        },
        {
            "key": "x",
            "label": "x",
            "format": "text"
        },
        {
            "key": "y",
            "label": "y",
            "format": "text"
        },
        {
            "key": "angle_deg",
            "label": "Acute angle between the lines",
            "format": "number"
        }
    ],
    "formula": {
        "text": "For a₁x + b₁y = c₁ and a₂x + b₂y = c₂:  D = a₁b₂ − a₂b₁;  x = (c₁b₂ − c₂b₁) ÷ D;  y = (a₁c₂ − a₂c₁) ÷ D;  tan θ = |D| ÷ |a₁a₂ + b₁b₂|.  y = mx + b is −mx + y = b.",
        "variables": {
            "a, b, c": "coefficients of each line",
            "m, b": "slope and y-intercept",
            "D": "determinant; D = 0 means parallel or identical lines",
            "θ": "acute angle between the lines"
        }
    },
    "worked_example": {
        "inputs": {
            "form": "slope",
            "m1": "2",
            "k1": "1",
            "m2": "-1",
            "k2": "4",
            "a1": "1",
            "b1": "1",
            "c1": "5",
            "a2": "1",
            "b2": "-1",
            "c2": "1"
        },
        "steps": [
            "Set the lines equal: 2x + 1 = −x + 4, so 3x = 3 and x = 1.",
            "Substitute: y = 2(1) + 1 = 3.",
            "Angle: tan θ = |(m₁ − m₂) ÷ (1 + m₁m₂)| = |3 ÷ (−1)| = 3, so θ ≈ 71.5651°."
        ],
        "result": "The lines y = 2x + 1 and y = −x + 4 cross at (1, 3), at an angle of about 71.57°."
    },
    "interpretation": "The intersection is the one (x, y) pair that satisfies both equations, so this is also a way to solve a pair of simultaneous linear equations.",
    "assumptions": [
        "Both lines lie in the same plane (2D)."
    ],
    "limitations": [
        "A vertical line (x = constant) cannot be written as y = mx + b; use the ax + by = c form with b = 0.",
        "Lines in 3D usually do not meet at all (they are skew); this tool covers the plane only."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How do I enter a vertical line such as x = 3?",
            "answer": "Choose ax + by = c and enter a = 1, b = 0, c = 3."
        },
        {
            "question": "When are two lines perpendicular?",
            "answer": "When m₁ × m₂ = −1, or in standard form when a₁a₂ + b₁b₂ = 0."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Slope Calculator",
            "url": "https://calculators.geekmidst.com/calculators/slope-calculator"
        },
        {
            "name": "Parabola Vertex Calculator",
            "url": "https://calculators.geekmidst.com/calculators/parabola-vertex-calculator"
        },
        {
            "name": "Distance Calculator",
            "url": "https://calculators.geekmidst.com/calculators/distance-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}