Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve a small linear program by checking the geometry of its constraints.
- Justify the conclusion "The maximum is 10 at (2,2)" using the stated assumptions.
Before you start
Linear inequalities and coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Maximize 3x+2y subject to x,y≥0, x+y≤4, and x≤2.
Why this math matters
Solve a small linear program by checking the geometry of its constraints. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The variables are continuous real quantities.
- The constraints are exactly those stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Locate an optimum among feasible polygon corners
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Maximize 3x+2y subject to x,y≥0, x+y≤4, and x≤2.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
The feasible vertices are (0,0),(2,0),(2,2),(0,4)
Intersections of active boundaries describe the polygon's corners.
Work through the mathematics
Objective values are 0,6,10,8
Evaluate the same linear objective at each candidate.
Check the conclusion
The maximum is 10 at (2,2)
A linear objective on this compact polygon attains a maximum at a vertex; here the best vertex is unique.
The result
The maximum is 10 at (2,2)
A linear objective on this compact polygon attains a maximum at a vertex; here the best vertex is unique.
Common mistakes to catch
- Integer restrictions would define a different optimization problem.
- A feasible corner must satisfy every inequality.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Maximize x+y on the same region.
Show a hint
Look along the boundary x+y=4.
Reveal answer and explanation
Maximum four, attained on the segment from (0,4) to (2,2)
Parallel objective lines can produce multiple optimizers.
Practice 2
Would checking vertices work without a bounded feasible region?
Show a hint
A maximum might not exist.
Reveal answer and explanation
Not without checking unbounded directions
An objective can increase indefinitely along a feasible ray.
Take the idea with you
Interpret the constraints as capacities in a two-product resource allocation model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compare distance rules and their unit balls
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