# Linear programming · Year 11

> Linear programming, Year 11: Regions of the plane and more. Explanation, worked examples and free practice for the national curriculum in England.

In Linear programming (Year 11) you practise 2 skills: Regions of the plane · The vertices of the region.

Linear programming looks for the best plan under limits. One variable stands for the number of items of the first kind and a second for the number of the second kind. Each limit in the story, such as hours of work or an amount of material, is an inequality, and neither variable is negative.

Each inequality describes a half plane: one side of a line. To tell which side, substitute the origin. The part shared by all the half planes is the region of possible plans, and its vertices are the points where two lines meet, or a line meets an axis.

The objective, such as the profit or the cost, reaches its greatest or least value at one of the vertices. Find every vertex, substitute each in the function and choose. When the constraints say “at least”, the region is open upwards, and the least cost is found there.

**Tip:** Before calculating, write in one sentence what each variable stands for and exactly what should be made greater or smaller.

## What you learn

### Regions of the plane

- Example: The inequality x + 2y ≥ 50 is given. Picture A: (Picture A: a line, its side with the origin shaded) Picture B: (Picture B: a line, its side away from the origin shaded) Which picture shows all the points that satisfy the inequality?
- Options: Picture A · Picture B
- Answer: Picture B
- Solution: Substitute the origin: 1 × 0 + 2 × 0 = 0 , which is less than 50, so the origin does not satisfy the inequality. So shade the side away from the origin. That is Picture B.

### The vertices of the region

- Example: The line 7x + 3y = 210 is given. At which point does it meet the x axis?
- Answer: (30, 0)
- Solution: Every point on the x axis has y = 0 . Substitute: 7x = 210 , so x = 30 . The point is (30, 0) .

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 11 topics](https://kixplo.com/uk/en/maths/year-11/)