Vertices & the Objective

The Corner as a Two by Two Solve

A fractional vertex is just the exact intersection of two binding lines. This lesson compares two displayed systems and the matching geometry beside them. The goal is to make a corner feel like a solved equation, not a guessed point.

highlighted = computed this step

Binding system

The fractional corner is where 2x+y=4 and x plus 2y=4 meet. Why: the matrix is just a compact record of the two boundary equations.

[214124]\begin{bmatrix}2&1&4\\1&2&4\end{bmatrix}
Binding equations and cornerEach displayed augmented matrix and polytope show the same exact corner.Exact rational matrix214124solution: 4/3, 4/3Feasible region2x + y ≤ 4x + 2y ≤ 4(0, 0)(0, 2)(4/3, 4/3)(2, 0)Exact rational matrix316124solution: 8/5, 6/5Feasible region3x + y ≤ 6x + 2y ≤ 4(0, 0)(0, 2)(8/5, 6/5)(2, 0)

Nonzero determinant

The determinant is 2 times 2 minus 1 times 1, which equals 3. Nonzero means the crossing is unique. Why: parallel boundaries would not produce a single vertex.

2⋅2−1⋅1=32\cdot{}2-{}1\cdot{}1=3
Binding equations and cornerEach displayed augmented matrix and polytope show the same exact corner.Exact rational matrix214124solution: 4/3, 4/3Feasible region2x + y ≤ 4x + 2y ≤ 4(0, 0)(0, 2)(4/3, 4/3)(2, 0)Exact rational matrix316124solution: 8/5, 6/5Feasible region3x + y ≤ 6x + 2y ≤ 4(0, 0)(0, 2)(8/5, 6/5)(2, 0)

Second solve

A second system uses 3x+y=6 with x plus 2y=4. Its determinant is 5, and the recomputed corner is (8/5, 6/5). Why: different slopes give different fractions, but the same exact solve certifies the point.

det⁡=5(x,y)=(8/5,6/5)\det=5\quad (x,y)=(8/5, 6/5)
Binding equations and cornerEach displayed augmented matrix and polytope show the same exact corner.Exact rational matrix214124solution: 4/3, 4/3Feasible region2x + y ≤ 4x + 2y ≤ 4(0, 0)(0, 2)(4/3, 4/3)(2, 0)Exact rational matrix316124solution: 8/5, 6/5Feasible region3x + y ≤ 6x + 2y ≤ 4(0, 0)(0, 2)(8/5, 6/5)(2, 0)

Exact corner

Solving the displayed system gives x=4/3 and y=4/3. The corner is (4/3, 4/3), exact, not a rounded decimal. Why: the diagram is allowed to round pixels, but not the values it displays.

x=4/3y=4/3(4/3,4/3)x=4/3\quad y=4/3\quad (4/3, 4/3)
Binding equations and cornerEach displayed augmented matrix and polytope show the same exact corner.Exact rational matrix214124solution: 4/3, 4/3Feasible region2x + y ≤ 4x + 2y ≤ 4(0, 0)(0, 2)(4/3, 4/3)(2, 0)Exact rational matrix316124solution: 8/5, 6/5Feasible region3x + y ≤ 6x + 2y ≤ 4(0, 0)(0, 2)(8/5, 6/5)(2, 0)

Diagram note

A fractional vertex is a boundary intersection verified by exact linear algebra. Pixel positions are rounded for layout; every number shown is exact.

solve the displayed equations, then plot the exact point\text{solve the displayed equations, then plot the exact point}
Binding equations and cornerEach displayed augmented matrix and polytope show the same exact corner.Exact rational matrix214124solution: 4/3, 4/3Feasible region2x + y ≤ 4x + 2y ≤ 4(0, 0)(0, 2)(4/3, 4/3)(2, 0)Exact rational matrix316124solution: 8/5, 6/5Feasible region3x + y ≤ 6x + 2y ≤ 4(0, 0)(0, 2)(8/5, 6/5)(2, 0)