Feasible Region and Corner Points

Lesson 4 of 10: Linear Programming

In this lesson:

• Name and locate the corner points of the feasible region
• Compute interior corner coordinates using simultaneous equations
• State the Fundamental Theorem and understand why only corners matter

By the End of This Lesson

In this lesson, you will:

1. Identify corners of F by tracing its boundary on the graph
2. Compute interior corner coordinates using simultaneous equations
3. Verify whether a candidate point lies on F's boundary
4. State the Fundamental Theorem and explain why only corners matter

Warm-Up: Solve These Equations First

Solve by substitution — you'll need this in Chunk C:

• Equation 1:
• Equation 2:

What values of and satisfy both equations simultaneously?

Show every step — this algebra drives today's main calculation.

F Is a Polygon — Name Its Corners

LP-03 ended with the feasible region F — a polygon with infinitely many valid plans.

• F is a polygon. Polygons have corners.
• Today: identify each corner, compute its coordinates, and discover why only corners matter

The LP-03 diagram is in front of you — look for the corners.

Trace the Boundary to Find Corners

Boundary = constraint lines + coordinate axes — every direction-change is a corner

Four Corners — Marked and Labelled

Each direction-change in the trace is now a labelled filled dot — these are the four corners of F

What Makes a Point a Corner of F?

Corner point — a point where two boundary lines meet on F's boundary.

• "On the boundary" separates a corner from two lines merely intersecting
• The trace gave us this — boundary kinks are corners by definition

Two lines can meet outside F — next chunk shows why.

Which Lines Form Each Corner?

Which two boundaries form each corner?

Boundaries: , , ,

Have We Found All the Corners?

Tracing found four corners — but tracing only visits intersections visible on the outline.

• Lines and must intersect somewhere — is that intersection a corner of F?
• How do we know we haven't missed candidates? We need a verification check for any intersection.

Verify the Candidate Point (6, 0)

• Step 1: Set in : . Mark (6, 0) with an open circle
• Step 2: Substitute : ✓, ✗ — fails here
• Step 3: One failure disqualifies — (6, 0) is not a corner of F.

The Verification Rule for Any Candidate

Every candidate corner must pass every constraint — one failure is enough to disqualify.

• Two boundary lines can meet outside F; the constraint check is what proves the candidate lies on F's actual boundary
• This is LP-03's substitute-and-check method — repurposed from test-point shading to corner verification

Your Turn: Verify the Point (0, 5)

• Lines and intersect at (0, 5) — a candidate corner of F
• Substitute into each constraint: , , ,
• Is a corner of F? Name the failing constraint.

Show all four substitutions

Axis Corners vs. Interior Corners

Axis corners: at least one boundary line is a coordinate axis. Coordinates are intercepts — readable off the graph.

Interior corners: formed by two non-axis constraint lines. Coordinates may be non-integer. Never trust visual inspection.

→ Rule: Axis corners — read off the graph. Interior corners — calculate every time.

Computing the Interior Corner: Steps 1–3

Corner C = (3, 2) is where and meet. The 5-step procedure:

Step 1: Identify the two boundary equations: and

Step 2: Isolate from the simpler equation:

Step 3: Substitute into the other:

Computing the Interior Corner: Steps 4–5

From the previous slide:

Step 4: Expand and solve:

Step 5: Back-substitute and verify:
. Check: ✓ and ✓

Corner C = (3, 2) confirmed.

The 5-Step Procedure — All at Once

Use this as your reference for any interior corner:

1. Identify the two boundary equations at that corner
2. Isolate (or ) from the simpler equation
3. Substitute into the other equation — one unknown remains
4. Solve for that unknown
5. Back-substitute and verify in both original equations

Guided: Axis Corner (5, 0) — Same Method

Corner B = (5, 0) sits on and . Apply the 5-step procedure:

• Step 2: is already isolated — the axis equation gives us this directly
• Steps 3–5: Substitute into : . Verify: ✓, ✓

Same five steps, one substitution

The LP-04 Deliverable: Corner-Point Table

The Fundamental Theorem of LP

Fundamental Theorem of LP — write this down:

For a linear objective function over a bounded feasible region, the maximum and minimum always occur at corner points.

• This reduces the infinite candidate set to four arithmetic evaluations — one per corner
• The contour-line visual next explains why

Level Lines Reveal the Winner

Slide a ruler outward from the origin — the last point where it touches F is always a corner

Why Only Corners Need Evaluating

From the visual: any interior point can be beaten — a parallel shift outward still intersects F.

• Without this theorem: infinitely many candidates to evaluate
• With this theorem: 4 arithmetic evaluations — one per corner

Checking interior points like (2, 2) is unnecessary — a corner always wins.

What Does the Contour-Line Diagram Show?

• From the diagram: which corner does touch last before leaving F?
• In one sentence: why must the maximum of a linear objective be at a corner, not an interior point?

Commit to both before advancing — state the second as the Theorem in your own words

Avoid These Four Common Mistakes

M1: Check ALL pairwise intersections — traced corners aren't the only candidates

M2: Interior corners may not be integers — use the 5-step method

M3: F is the polygon of kinks — not the bounding rectangle

M4: The theorem makes interior-point checks redundant — trust it

LP-04 Delivers the Corner-Point Table

✓ Corners of F: identified by tracing, verified by substitution, computed exactly

✓ LP-04 deliverable: corners A(0,0), B(5,0), C(3,2), D(0,4) — the Fundamental Theorem reduces optimisation to these four

✓ LP-05: add an objective column, evaluate each row, pick the maximum

Four evaluations, one decision. Next lesson, we make it.