GCSE Maths: Linear Programming – Revision Guide | GCSE 数学：线性规划考点精讲

📚 GCSE Maths: Linear Programming – Revision Guide | GCSE 数学：线性规划考点精讲

Linear programming is a powerful mathematical technique used to find the best outcome (such as maximum profit or minimum cost) in a given model. For GCSE Maths, you need to understand how to translate real-world constraints into inequalities, graph them to obtain a feasible region, and then use an objective function to locate the optimal solution. Mastering this topic not only helps with exam questions but also develops logical reasoning skills.

线性规划是一种强大的数学方法，用于在给定的模型中寻找最佳结果（如最大利润或最低成本）。在 GCSE 数学中，你需要学会如何将现实约束转化为不等式，绘制它们以获得可行域，然后利用目标函数找到最优解。掌握这一考点不仅有助于应对考试，还能培养逻辑推理能力。

1. What is Linear Programming? | 什么是线性规划？

Linear programming (LP) deals with optimising a linear objective function subject to a set of linear inequalities called constraints. The variables, typically x and y, cannot be negative in most practical problems. The goal is to find values of x and y that maximise or minimise the objective while satisfying all constraints.

线性规划处理的是在一组线性不等式（称为约束条件）下优化一个线性目标函数的问题。变量通常为 x 和 y，在大多数实际问题中不能为负值。目标是找到满足所有约束条件并使目标最大化或最小化的 x 和 y 值。

In GCSE, problems often involve two variables and are solved graphically. This visual approach makes it easier to understand and verify solutions.

在 GCSE 中，问题通常涉及两个变量并用图形方法求解。这种直观的方法便于理解和验证解。

2. Translating Word Problems into Inequalities | 将应用题转化为不等式

The first step is to identify the decision variables. For example, let x be the number of product A and y be the number of product B. Then, read each constraint and express it as an inequality. Typical phrases: ‘at most’ means ≤, ‘at least’ means ≥, ‘cannot exceed’ means ≤, and ‘must be at least’ means ≥.

第一步是确定决策变量。例如，设 x 为产品 A 的数量，y 为产品 B 的数量。然后，阅读每个约束条件并用不等式表示。典型表述：”最多” 表示 ≤，”至少” 表示 ≥，”不能超过” 表示 ≤，”必须至少” 表示 ≥。

Also, non-negativity constraints x ≥ 0, y ≥ 0 are almost always implied unless stated otherwise. Always list them.

另外，除非另有说明，非负约束 x ≥ 0, y ≥ 0 几乎总是隐含的。务必列出它们。

3. Drawing Boundaries: Equations and Lines | 绘制边界：方程与直线

For each inequality, first replace the inequality sign with an equals sign to get the boundary line equation. For example, 2x + y ≤ 10 becomes 2x + y = 10. Plot this straight line by finding two points (commonly x- and y-intercepts). Use a solid line if the inequality includes equality (≤ or ≥); use a dashed line if it is strict (< or >).

对于每个不等式，先将不等号替换为等号得到边界直线方程。例如，2x + y ≤ 10 变为 2x + y = 10。通过找两点（通常是 x 轴和 y 轴截距）绘制这条直线。如果不等式包含等号（≤ 或 ≥），使用实线；如果是严格不等式（< 或 >），使用虚线。

Always label your lines clearly. Axes should be marked with appropriate scales so that the feasible region fits nicely on the graph paper.

始终清晰地标注直线。坐标轴应标有合适的刻度，以便可行域能很好地展现在图纸上。

4. Shading the Correct Side of an Inequality | 为不等式正确区域涂阴影

To determine which side of the line to shade, choose a test point not on the line, with (0,0) being the easiest if the line does not pass through the origin. Substitute the coordinates into the original inequality. If the statement holds true, shade the side containing the test point; otherwise, shade the opposite side.

要确定直线的哪一侧涂阴影，选择一个不在直线上的测试点，若直线不经过原点，(0,0) 是最简单的测试点。将坐标代入原不等式。如果不等式成立，则涂包含测试点的一侧；否则涂另一侧。

In many GCSE problems, you are asked to leave the feasible region unshaded and shade the rejected regions. Check the question’s instruction carefully. Whichever method is used, the feasible region will be the area where all constraints are satisfied simultaneously.

在许多 GCSE 题目中，要求留出可行域不涂阴影，而涂掉不可行区域。仔细阅读题目要求。无论用哪种方法，可行域都是同时满足所有约束条件的区域。

5. The Feasible Region | 可行域

After shading all inequalities according to the rules, the feasible region is the intersection of all allowed areas. It is a polygon (often a triangle or quadrilateral) whose vertices are the points of intersection of the boundary lines. Mark these vertices clearly or list their coordinates.

按照规则为所有不等式涂阴影后，可行域是所有允许区域的交集。它是一个多边形（通常是三角形或四边形），其顶点是边界线的交点。清楚地标记这些顶点或列出它们的坐标。

If the feasible region is unbounded (open on one side), the question will typically still ask for a maximum or minimum that occurs at a vertex.

如果可行域是无界的（一侧开放），题目通常仍会要求找出在某个顶点处取得的最大值或最小值。

6. Introducing the Objective Function | 引入目标函数

The objective function expresses the quantity to be optimised. For instance, P = 3x + 5y represents the total profit from selling x units of item A and y units of item B, where profits are £3 and £5 per unit respectively. We aim to maximise P or minimise cost C = 20x + 30y.

目标函数表达了需要优化的量。例如，P = 3x + 5y 表示销售 x 个 A 产品和 y 个 B 产品的总利润，单位利润分别为

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