📚 GCSE CCEA Maths: Linear Programming – Key Points Explained | GCSE CCEA 数学：线性规划 考点精讲
Linear programming is a powerful tool in GCSE CCEA Mathematics that allows you to find the optimal value—such as maximum profit or minimum cost—within a set of real-world limitations. By using graphs and algebraic methods, you can solve problems involving two variables and several linear constraints. Mastering this topic means learning to translate worded scenarios into inequalities, shade feasible regions accurately, and use the vertex method to obtain the best solution. This article breaks down every key concept, common pitfalls, and exam-style skills you need to excel.
线性规划是 GCSE CCEA 数学中一个强大的工具，它能让你在现实限制条件中找到最优值——例如最大利润或最小成本。通过使用图表和代数方法，你可以解决涉及两个变量和若干线性约束的问题。掌握这一主题意味着学会将文字情景转化为不等式、准确地涂出可行区域，并运用顶点法求得最佳解。本文将逐一剖析每个关键概念、常见陷阱以及你需要掌握的考试技能。
1. Introduction to Linear Programming | 线性规划简介
Linear programming is a mathematical technique designed to optimise (maximise or minimise) a linear objective function subject to a set of linear constraints. At the GCSE level on the CCEA specification, problems are always restricted to two variables, typically x and y, making it possible to solve them graphically. The core idea is that the best possible outcome will occur at one of the corner points, or vertices, of the feasible region formed by the constraints.
线性规划是一种数学技术，旨在在一组线性约束条件下优化（最大化或最小化）一个线性目标函数。在 CCEA 考试局 GCSE 阶段，问题始终限制在两个变量，通常为 x 和 y，因此可以通过图形方式求解。其核心思想是，最佳可能结果将出现在由约束条件形成的可行区域的某个角点（即顶点）处。
2. Formulating the Problem | 建立数学模型
nario into precise mathematical language. Start by defining your decision variables: write 'Let x be the number of ...' and 'Let y be the number of ...'. Every limiting condition must become a linear inequality. For example, 'the total number of packages cannot exceed 80' s x ≥ 0 and y ≥ 0, because quantities like time, items, or ingredients cannot be negative.
第一个关键技能是将现实情景转换为精确的数学语言。首先要定义决策变量：写出“设 x 为……的数量”，“设 y 为……的数量”。每个限制条件都必须转化为一个线性不等式。例如，“包裹总数不能超过 80” 转化为 x + y ≤ 80。不要忘记非负约束 x ≥ 0 和 y ≥ 0，因为时间、物品或原料等数量不可能是负数。
3. Defining Constraints | 定义约束条件
inear programming problem. They may come from resources, budgets, production hours, or physical limits. CCEA questions often use phrases like 'at most', 'no more than', and 'at least'. Be careful when translating comparative statements: 'the number of deluxe boxes is at least three times the number of standard boxes' lways check the direction of your inequality by substituting a small test number mentally.
约束条件是任何线性规划问题的支柱。它们可能来源于资源、预算、生产工时或物理限制。CCEA 的题目中经常使用“至多”、“不超过”和“至少”等词语。转化比较性陈述时要小心：比如“豪华礼盒的数量至少是标准礼盒的三倍”应转化为 y ≥ 3x，而不是 3y ≥ x。可以通过在心中代入一个小的测试数字来检查不等式的方向是否正确。
4. Graphing Inequalities | 绘制不等式区域
s included, and a dashed line for < or > when the boundary itself is not part of the solution. Then pick a test point—usually (0,0) if it is not on the line— e the unwanted regions, leaving the feasible region clear. Always read the question carefully to see which side they want shaded.
要绘制线性不等式的图形，首先将其边界线视为方程来绘制。在 CCEA 的网格纸上，对于 ≤ 或 ≥ 使用实线以表示该线包含在区域内，对于 < 或 > 则使用虚线，因为边界本身不属于解集。然后选取一个测试点——通常用 (0,0) 如果它不在线上——并检查该点是否满足不等式。将不满足约束条件的区域涂上阴影。许多 CCEA 试卷的要求是将不需要的区域涂上阴影，以便清晰地保留可行区域。务必仔细读题，看清楚他们要求对哪一侧涂阴影。
5. Identifying the Feasible Region | 识别可行区域
Once all inequalities are drawn, the feasible region is the intersection of all the allowed half-planes—the part of the graph that remains unshaded (or occasionally lightly shaded as the target region). This region is a convex polygon (or possibly unbounded). Label the vertices clearly, as they will be used to find the optimal solution. Two boundary lines intersect at a vertex; you can find its coordinates by solving the simultaneous equations of those two lines.
当所有不等式绘制完成后，可行区域就是所有允许的半平面的交集——即图形中未被涂上阴影的部分（或偶尔被浅色标记为目标区域）。这个区域是一个凸多边形（或者可能是无界区域）。清晰地标出顶点，因为它们将被用来寻找最优解。两条边界线相交于一个顶点；你可以通过求解这两条直线的方程组来找到其坐标。
6. Objective Function and Optimal Solutions | 目标函数与最优解
ding to the corner-point theorem, for a bounded feasible region the optimal value will always be found at one of the vertices. Therefore, you do not need to test random points inside the region; simply evaluate the objective function at each vertex. In a maximisation problem, pick the largest value; for minimisation, pick the smallest. Always express your final answer in terms of the original context, stating the number of each item and the total profit or cost.
目标函数是你试图最大化或最小化的表达式，例如利润 P = 5x + 7y。根据角点定理，对于有界的可行区域，最优值总会在其中一个顶点处找到。因此，你不需要测试区域内的随机点；只需在每个顶点处评估目标函数。在最大化问题中，选取最大值；在最小化问题中，选取最小值。始终将最终答案还原到原始情景中，说明每种物品的数量以及总利润或总成本。
7. Vertex Method for Maximisation | 最大化问题的顶点法
able. (ii) Write down the coordinates of each vertex. (iii) Substitute each pair into the objective function, e.g. P = 2x + 3y, and calculate the result. (iv) The vertex giving the highest value is the optimal solution. Show your substitution clearly so examiners can follow your reasoning. For instance, if vertices are (0,0), (0,40), (30,10), (45,0), you might calculate P values of 0, 120, 90, and 90, making (0,40) the optimum.
以下是最大化问题的系统步骤：(i) 确定可行区域的所有顶点，包括适用的坐标轴截距。(ii) 写出每个顶点的坐标。(iii) 将每个顶点代入目标函数，例如 P = 2x + 3y，并计算结果。(iv) 给出最高值的顶点即为最优解。清晰地展示你的代入过程，以便考官能跟上你的推理。例如，如果顶点为 (0,0)、(0,40)、(30,10)、(45,0)，你可以计算出 P 值分别为 0、120、90、90，从而使 (0,40) 成为最优点。
8. Minimisation Problems | 最小化问题
Minimisation follows the same vertex method, but you look for the smallest objective function value. Keep in mind that if the feasible region is unbounded in the direction of decreasing cost, a finite minimum may not exist; however, CCEA problems are nearly always constructed with a closed, bounded polygon to ensure a clear answer. Check that the coordinates of the minimum vertex satisfy all the original inequalities, and that they make practical sense—for example, you cannot run a machine for negative hours.
最小化问题遵循相同的顶点法，但你要寻找目标函数的最小值。请记住，如果可行区域在成本下降的方向上是无界的，那么有限的最小值可能不存在；不过，CCEA 的题目几乎都构建了一个封闭且有界的多边形，以确保有明确的答案。检查最小顶点的坐标是否满足所有原始不等式，并确保它们在实际中是有意义的——例如，你不能让机器运行负小时数。
9. Integer Solutions and Practical Context | 整数解与实际应用
When variables represent countable items — tables, people, books — the solution must be a whole number. If the optimal vertex has coordinates like (12.5, 27.3), you must search the integer points near that vertex. Mark all integer coordinate points within the feasible region, then test the objective function at a few of the closest feasible integer points. Be methodical: sometimes the integer optimum is not simply the rounded coordinates, so you might need to test (12,27), (13,27), (12,28), (13,28) etc. CCEA expects you to show this checking process and state the final integer solution clearly.
当变量代表可计数的物品——桌子、人、书本——时，解必须是整数。如果最优顶点的坐标例如是 (12.5, 27.3)，你就需要在该顶点附近的整数点中搜寻。标记出可行区域内的所有整数坐标点，然后在距离最优顶点最近的几个可行整数点处测试目标函数。要有条不紊：有时整数最优解并不仅仅是四舍五入后的坐标，因此你可能需要测试 (12,27)、(13,27)、(12,28)、(13,28) 等点。CCEA 要求你展示这一检查过程，并清晰地陈述最终的整数解。
10. Common Exam Pitfalls | 常见考试陷阱
repeating these errors: (1) Not defining variables explicitly at the start. (2) Confusing 'at least' (≥) with 'at most' cause the test point was not checked. (5) Assuming the optimum lies where two specific lines intersect without testing all vertices. (6) Forgetting to verify that the chosen vertex satisfies all constraints, especially after finding an integer solution. (7) Failing to answer the question in context— le, stating only coordinates rather than the numbers of each product and the total profit.
即使成绩好的考生也会因重复这些错误而丢分：(1) 一开始未明确定义变量。(2) 混淆了“至少” (≥) 和“至多” (≤)。(3) 忘记对严格不等式如 < 或 > 使用虚线，或者使用错误。(4) 由于没有检查测试点，将边界线的错误一侧涂上了阴影。(5) 假定最优解出现在某两条特定直线的交点处，而没有测试所有顶点。(6) 忘记验证所选顶点满足所有约束条件，尤其是在找到整数解之后。(7) 未能结合情境回答问题——例如，只陈述了坐标，而没有给出每种产品的数量和总利润。
11. Worked Example: A Two-Variable Problem | 实例解析：双变量问题
≤ 50; material supply 2x + y ≤ 80; and x ≥ 0, y ≥ 0. The profit from a Standard hamper is £3 and from a Premium hamper is £4. Formulate and solve the linear programming problem to find the maximum profit.
一家公司生产两种礼篮，标准礼篮 (x) 和高级礼篮 (y)。约束条件为：包装能力 x + y ≤ 50；材料供应 2x + y ≤ 80；且 x ≥ 0, y ≥ 0。每个标准礼篮的利润为 3 英镑，每个高级礼篮的利润为 4 英镑。建立并求解该线性规划问题，以求得最大利润。
e 2x + y = 80 has intercepts (40,0) and (0,80). The non-negativity limits the region to the first quadrant. The feasible region is the polygon with vertices where these lines and axes intersect.
首先，确定目标函数：利润 P = 3x + 4y。接着，我们绘制约束条件。直线 x + y = 50 的截距为 (50,0) 和 (0,50)。直线 2x + y = 80 的截距为 (40,0) 和 (0,80)。非负约束将区域限制在第一象限。可行区域是由这些直线与坐标轴相交所形成的多边形。
The vertices of the feasible region are: (0,0) — the origin; (0,50) — where x=0 meets x+y=50; (40,0) — x = 30, then substitute into x+y=50 to find y = 20. So the intersection vertex is (30,20).
可行区域的顶点为：(0,0) — 原点；(0,50) — x=0 与 x+y=50 的交点；(40,0) — y=0 与 2x+y=80 的交点；以及 x+y=50 和 2x+y=80 的交点。联立求解：用第二个方程减去第一个方程，得到 x = 30，然后代入 x+y=50 求出 y = 20。因此交点顶点为 (30,20)。
Now evaluate the profit function at each vertex using a table to stay organised:
现在通过表格系统地评估每个顶点处的利润函数：
| Vertex (x, y) | P = 3x + 4y | Profit (£) |
|---|---|---|
| (0, 0) | 3(0) + 4(0) = 0 | 0 |
| (0, 50) | 3(0) + 4(50) = 200 | 200 |
| (40, 0) | 3(40) + 4(0) = 120 | 120 |
| (30, 20) | 3(30) + 4(20) = 170 | 170 |
The maximum profit is £200
Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
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