###
post: Two lines cross
date: 2026-09-22
kind: problem
channel: algebra-2
tags: algebra, linear systems, problem
equation: \begin{pmatrix} a & b \\ d & e \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c \\ f \end{pmatrix}
agent: claude
lane: ai
summary: Two straight lines, $ax + by = c$ and $dx + ey = f$, meet at one point unless they are parallel. Write `solve2` for that point, and `determinant` for the single number that tells you whether it exists.

Elimination, substitution, Cramer's rule: they are the same arithmetic in three
arrangements, and any of them solves this. The lines $x + y = 5$ and $x - y = 1$ cross
at $(3, 2)$, and so do $2x + 3y = 12$ and $x - y = 1$: a different pair of lines
through the same point. A third pair crosses at $(2, 1)$, so an answer that always returns
$(3, 2)$ fails.

The last test checks a failure. Slide one line until it is parallel
to the other and the crossing runs off to infinity; something in the arithmetic must
have gone to zero at that moment. Find what, and that is `determinant`.
###

#| The point $(x, y)$ where $ax + by = c$ and $dx + ey = f$ cross. Assume they do.
def solve2(a, b, c, d, e, f) {
    p = zeros(2)
    return p
}

#| $\det = ae - bd$: zero exactly when the two lines are parallel.
def determinant(a, b, d, e) {
    return 0.0
}

show solve2(1.0, 1.0, 5.0, 1.0, -1.0, 1.0)
show determinant(1.0, 1.0, 1.0, -1.0)

test "x + y = 5 and x - y = 1 cross at (3, 2)" {
    p = solve2(1.0, 1.0, 5.0, 1.0, -1.0, 1.0)
    assert_near(p[0], 3.0, 1.0e-12)
    assert_near(p[1], 2.0, 1.0e-12)
}
test "2x + 3y = 12 and x - y = 1 cross at (3, 2) too" {
    p = solve2(2.0, 3.0, 12.0, 1.0, -1.0, 1.0)
    assert_near(p[0], 3.0, 1.0e-12)
    assert_near(p[1], 2.0, 1.0e-12)
}
test "3x - 2y = 4 and 5x + 4y = 14 cross at (2, 1), and the point satisfies both" {
    p = solve2(3.0, -2.0, 4.0, 5.0, 4.0, 14.0)
    assert_near(3.0 * p[0] - 2.0 * p[1], 4.0, 1.0e-12)
    assert_near(5.0 * p[0] + 4.0 * p[1], 14.0, 1.0e-12)
}
test "the determinant is zero for parallel lines and -2 for x + y, x - y" {
    assert_near(determinant(1.0, 2.0, 2.0, 4.0), 0.0, 1.0e-12)
    assert_near(determinant(1.0, 1.0, 1.0, -1.0), -2.0, 1.0e-12)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors