Linear Programming for GCSE WJEC Maths | GCSE WJEC 数学：线性规划考点精讲

📚 Linear Programming for GCSE WJEC Maths | GCSE WJEC 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 situation where resources are limited. In the GCSE WJEC Mathematics specification, linear programming appears in the Higher tier, challenging students to translate real-world constraints into inequalities, graph feasible regions, and then determine the optimal solution using a methodical approach. Mastering this topic can earn you valuable marks in the exam. This guide breaks down every essential concept and skill you need, with clear dual-language explanations to help both English and Chinese learners.

线性规划是一种强大的数学方法，用于在资源有限的情况下找到最佳结果——例如最大利润或最低成本。在 GCSE WJEC 数学考试中，线性规划属于高阶段内容，要求学生将现实约束转化为不等式，画出可行域，然后用系统的方法确定最优解。掌握这个专题可以为你在考试中赢得宝贵的分数。本指南将逐项解析你需要的每个核心概念与技巧，并使用清晰的双语解释帮助英语和中文学习者。

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

Linear programming (LP) involves maximising or minimising a linear objective function subject to a set of linear constraints. The constraints are expressed as linear inequalities, and the variables (usually x and y) represent quantities that cannot be negative. For example, a company might want to maximise its weekly profit (P) by producing two products, but its production is limited by available machine hours and materials. The solution is found at one of the vertices (corner points) of the feasible region formed by the intersection of all constraints. This geometrical approach is what you will practise in your WJEC exam.

线性规划涉及在一组线性约束条件下，最大化或最小化一个线性目标函数。约束条件表现为线性不等式，变量（通常用 x 和 y 表示）代表不能为负的数量。例如，一家公司可能希望最大化其每周利润 (P)，通过生产两种产品，但生产受到可用机器工时和原材料的限制。最优解出现在所有约束条件交集形成的可行区域的一个顶点（角点）上。这种几何方法正是你在 WJEC 考试中需要练习的。

2. Defining the Variables and Identifying Constraints | 定义变量与识别约束

The first step in any linear programming problem is to clearly define the decision variables. For instance, let x = number of Type A items produced, and y = number of Type B items. Every LP problem also includes a key constraint: x ≥ 0 and y ≥ 0, because quantities cannot be negative. Then, read the problem carefully to extract linear inequalities representing resource limits. For example, if each Type A requires 2 hours and each Type B requires 3 hours, and total hours available are 120, the inequality is 2x + 3y ≤ 120. Always express constraints in simplest integer form.

任何线性规划问题的第一步都是清晰定义决策变量。例如，设 x = 生产A类产品的数量，y = 生产B类产品的数量。每个线性规划问题还包括一个关键约束：x ≥ 0 且 y ≥ 0，因为数量不能为负。然后，仔细阅读题目，提取代表资源限制的线性不等式。例如，如果每个A类产品需要2小时，每个B类产品需要3小时，而总可用工时为120，则不等式为 2x + 3y ≤ 120。始终将约束表示为最简整数形式。

3. Translating Worded Constraints into Inequalities | 把文字约束转化为不等式

WJEC exam questions often describe constraints in words. Key phrases include: ‘at least’ means ≥, ‘at most’ means ≤, ‘no more than’ is ≤, ‘minimum’ is ≥, and ‘must not exceed’ is ≤. When a constraint combines two products, pay attention to the coefficients (the amounts per item). For example, ‘Each Type A uses 5 kg of material, each Type B uses 4 kg; total material available is 80 kg’ becomes 5x + 4y ≤ 80. Also, a typical constraint might be a ratio or a minimum production requirement, such as ‘the total number of items must be at least 30’ → x + y ≥ 30.

WJEC 考试题常用文字描述约束。关键短语包括：“至少”表示 ≥，“至多”表示 ≤，“不超过”为 ≤，“最小值”为 ≥，“不得超过”为 ≤。当约束条件涉及两种产品时，注意系数（每件产品的用量）。例如，“每个A类产品用5公斤材料，每个B类用4公斤；总材料有80公斤”转化为 5x + 4y ≤ 80。另外，常见约束可能是比例或最低生产要求，如“产品总数至少为30” → x + y ≥ 30。

4. Graphing Linear Inequalities | 绘制线性不等式

To graph an inequality like 2x + 3y ≤ 120, first draw the boundary line 2x + 3y = 120 as a solid line (since ≤ includes the equality). For strict inequalities (< or >), use a dashed line. Find the intercepts: when x=0, y=40; when y=0, x=60. Plot these points and connect. Then choose a test point not on the line, typically (0,0). Substitute into the inequality: 2(0)+3(0) ≤ 120 → 0 ≤ 120, true, so shade the region containing (0,0). Repeat for all constraints, and the overlapping region is the feasible region. In WJEC, you may need to shade the unwanted regions and leave the feasible region unshaded, or shade the feasible region directly – follow the question’s instruction.

要画出不等式如 2x + 3y ≤ 120 的图像，首先画出边界直线 2x + 3y = 120，用实线（因为≤包含等号）。对于严格不等式（< 或 >），用虚线。找出截距：当 x=0 时，y=40；当 y=0 时，x=60。描点连线。然后选取不在线上的测试点，通常用(0,0)。代入不等式：2(0)+3(0) ≤ 120 → 0 ≤ 120，成立，因此涂阴包含(0,0)的区域。对所有约束重复此步，重叠区域即为可行域。在 WJEC 考试中，你可能需要涂掉不满足的区域，保留可行区域不涂，或者直接涂阴可行域——请遵循题目指示。

5. Identifying the Feasible Region | 识别可行域

The feasible region is the set of all points that satisfy every inequality simultaneously. Graphically, it is the common overlap of all shaded areas. This region is often a convex polygon bounded by the constraint lines and the axes. The vertices (corner points) of this polygon are critical because they will be tested to find the optimal solution. Label these vertices clearly on your graph, either by reading coordinates directly from the graph or by solving the corresponding pair of equations. Inaccuracies in drawing can lead to wrong coordinates, so whenever possible, solve algebraically for the exact intersection points of two lines.

可行域是所有同时满足每个不等式的点的集合。在图上，它是所有涂阴区域的公共重叠部分。这个区域通常是由约束线和坐标轴围成的一个凸多边形。该多边形的顶点（角点）至关重要，因为我们将通过检验它们来找到最优解。在图上清晰地标注这些顶点，可以直接从图上读坐标，也可以通过解对应的方程求得。绘图不准确可能导致坐标错误，因此，只要可能，都应通过代数方法精确求解两条直线的交点坐标。

6. Understanding the Objective Function | 理解目标函数

The objective function expresses the quantity to be maximised or minimised, often profit or cost. It has the form F = ax + by, where F is the total value, and a, b are given constants. For example, if each Type A gives a profit of £6 and each Type B gives £8, the objective is maximise P = 6x + 8yPublished by TutorHao | GCSE Mathematics Revision Series | aleveler.com