IGCSE WJEC Maths: Linear Programming – Essential Revision | 线性规划考点精讲

📚 IGCSE WJEC Maths: Linear Programming – Essential Revision | 线性规划考点精讲

Linear programming is a powerful tool for solving optimisation problems where you need to maximise or minimise a linear objective function subject to a set of linear inequalities. For IGCSE WJEC Maths, you are expected to formulate constraints, graph feasible regions, and find optimal solutions – often in the context of real-world scenarios like profit, cost, or resource allocation. This revision guide covers all the key skills you need to ace this topic, with exam-focused explanations and bilingual support.

线性规划是解决优化问题的有力工具，旨在满足一组线性不等式约束的条件下，最大化或最小化一个线性目标函数。在 IGCSE WJEC 数学考试中，你需要能够建立约束条件、绘制可行域并寻找最优解——通常应用于利润、成本或资源分配等现实情境。本考点精讲将涵盖你需要掌握的所有关键技能，并提供考试导向的双语讲解，助你轻松应考。

1. Understanding Linear Programming | 理解线性规划

Linear programming is a mathematical technique used to find the best outcome – such as maximum profit or minimum cost – in a model whose requirements are represented by linear relationships. At IGCSE level, you will only work with two variables, typically x and y. The power of the method lies in its ability to handle multiple constraints simultaneously and identify an optimal decision from countless possibilities.

线性规划是一种数学技术，用于在需求由线性关系表示的模型中寻找最佳结果，例如最大利润或最小成本。在 IGCSE 阶段，你只需要处理两个变量，通常是 x 和 y。该方法的能力在于能同时处理多个约束条件，并从无数可能性中找出最优决策。

Every linear programming problem involves two core components: a set of linear inequalities called constraints, and a linear expression to be maximised or minimised, called the objective function. For WJEC exams, the context is often practical – you might be asked to plan production of two products, schedule workers, or allocate limited resources.

每个线性规划问题都包含两个核心部分：一组称为约束条件的线性不等式，以及一个需要最大化或最小化的线性表达式，即目标函数。在 WJEC 考试中，题目情境通常是实际的——你可能需要规划两种产品的生产、安排工人或分配有限资源。

Succeeding in this topic requires a clear understanding of how to translate word problems into inequalities, how to accurately graph them, and how to extract the optimal solution by testing key points. It is one of the most heavily weighted topics in the AO3 problem-solving strand.

要在这个主题中取得成功，你需要清楚理解如何将文字问题转化为不等式，如何准确绘制其图形，以及如何通过测试关键点得出最优解。这是 AO3 问题解决模块中权重很高的考点之一。

2. Formulating Constraints as Inequalities | 将约束条件表示为不等式

The first step in any linear programming problem is to define your variables clearly. For instance, “let x be the number of chairs produced and y be the number of tables produced.” All constraints must then be written as linear inequalities in terms of these variables. Typical constraints include limits on total quantity, time, materials, or budget.

任何线性规划问题的第一步是清晰定义变量。例如，“设 x 为生产的椅子数量，y 为生产的桌子数量。”然后所有约束条件都必须用这些变量写为线性不等式。常见的约束包括总量限制、时间、材料或预算。

The table below shows how common worded restrictions translate into mathematical form:

下表展示了常见的文字限制如何转化为数学形式：

Verbal Constraint (English)	约束描述（中文）	Inequality
At most 40 items in total	总共最多 40 件	x + y ≤ 40
At least 10 chairs	至少 10 把椅子	x ≥ 10
Twice as many chairs as tables, at most	椅子数量至多是桌子两倍	x ≤ 2y
Total labour hours: 2 per chair, 3 per table, max 120 hours	总工时：每椅2小时，每桌3小时，最多120小时	2x + 3y ≤ 120

Always remember to include the non-negativity constraints x ≥ 0 and y ≥ 0 unless the context explicitly says otherwise. These ensure that your solutions make sense in the real world, as you cannot produce a negative number of items.

务必记住加入非负约束 x ≥ 0 和 y ≥ 0，除非题目明确说明可以取负。这些约束确保解在现实中合理，因为你不可能生产负数量的物品。

Poorly defined variables or missing constraints are a frequent source of lost marks. Underline key numbers in the question and write each constraint one by one. Once all inequalities are listed, check them against the wording to avoid reversal of inequality signs.

变量定义不清或遗漏约束是常见失分点。划出题目中的关键数字，逐一写出每个约束。列出所有不等式后，对照文字检查，避免不等号方向颠倒。

3. Graphing Inequalities Correctly | 正确绘制不等式图形

Each linear inequality represents a half-plane on the coordinate grid. To graph it, first draw the boundary line as if it were an equation. Use a solid line for inequalities that include equality (≤ or ≥) and a dashed line for strict inequalities (< or >). This distinction is critical; many WJEC mark schemes award a mark specifically for using the correct line style.

每个线性不等式表示坐标平面上的一个半平面。要绘制它，首先像方程一样画出边界线。对于包含等号的不等式（≤ 或 ≥）使用实线，对于严格不等式（< 或 >）使用虚线。这一区分非常关键；WJEC 的许多评分方案会专门为使用正确的线型设分。

After drawing the line, pick a test point not on the line – usually (0,0) is convenient, provided it is not on the boundary. Substitute the coordinates into the inequality. If the inequality holds true, shade the side containing the test point; if false, shade the opposite side. Consistency is important: in WJEC exams you are typically asked to shade the unwanted region, leaving the feasible region unshaded. Always read the question instruction carefully.

画出直线后，选择一个不在直线上的测试点——通常取 (0,0) 很方便，只要它不在边界上。将坐标代入不等式。如果不等式成立，则涂阴影在包含测试点的一侧；如果不成立，则涂阴影在对侧。保持一致很重要：在 WJEC 考试中，通常要求对不符合区域涂阴影，从而留下空白的可行域。务必仔细阅读题目指令。

Use a ruler and a sharp pencil, and label your axes and lines clearly. Imprecise graphs can lead to incorrect vertices and consequently wrong optimal values. Plot at least two points accurately for each boundary line, and extend the lines far enough to see intersections clearly.

使用直尺和削尖的铅笔，并清晰地标注坐标轴和直线。不精确的图形可能导致错误的顶点坐标，进而得出错误的最优值。每条边界线至少精确绘制两个点，并将直线延伸足够远，以便清晰地观察交点。

4. Identifying the Feasible Region | 识别可行域

The feasible region is the area that satisfies all constraints simultaneously. On your graph, after shading the unwanted sides of every inequality, the clear unshaded polygon (or sometimes an unbounded region) is the feasible region. Every point inside or on the boundary of this region represents a possible solution that meets all restrictions.

可行域是同时满足所有约束条件的区域。在图上，对每个不等式的不符合侧涂阴影后，留下的清晰未涂阴影的多边形（有时为无界区域）就是可行域。该区域内或边界上的每一点都代表一个满足所有限制的可能解。

It is vital to label the feasible region with a capital R or simply indicate it. In an exam, you might be asked to find the coordinates of the vertices of this region. Read these directly from your graph where the boundary lines intersect. Sometimes you may need to solve two line equations simultaneously to get exact coordinates, especially if the intersection does not land on grid points.

用大写字母 R 标注可行域或直接标示出来至关重要。考试中，你可能会被要求找出该区域顶点的坐标。直接从图上读取边界线相交处的坐标。有时你可能需要联立求解两条直线的方程以获得精确坐标，尤其是当交点不在网格点上时。

If the region is unbounded (e.g., only two sides are fixed and it extends infinitely), the optimal value might still exist but you need to test carefully. For WJEC IGCSE, most problems produce a closed polygon such as a triangle or quadrilateral, making vertex testing straightforward.

如果区域是无界的（例如只有两条边是固定的并且区域无限延伸），最优值可能依然存在，但需要仔细测试。对于 WJEC IGCSE，大多数问题会产生一个封闭多边形，如三角形或四边形，使得顶点测试简单明了。

5. Writing the Objective Function | 写出目标函数

The objective function is the linear expression you are aiming to maximise or minimise. It usually represents profit, cost, time, or another quantity that depends on the variables x and y. A typical objective function might look like P = 40x + 30y, where each chair gives a profit of £40 and each table gives £30.

目标函数是你想要最大化或最小化的线性表达式。它通常表示利润、成本、时间或其他依赖于变量 x 和 y 的量。一个典型的目标函数可能形如 P = 40x + 30y，表示每把椅子利润 40 英镑，每张桌子利润 30 英镑。

The variables in the objective function must match the definitions used in the constraints. Carefully read whether the coefficient is per unit or per batch. Sometimes the problem asks for maximum income, minimum cost, or maximum number of items. Your objective function must directly reflect the wording.

目标函数中的变量必须与约束条件中使用的定义匹配。仔细阅读系数是每单位还是每批次的。有时题目要求的是最大收入、最小成本或最大物品数量。你的目标函数必须直接反映文字描述。

A common pitfall is confusing x and y roles. Double-check: if x is chairs and y is tables, then 40x + 30y means £40 per chair and £30 per table. Reversing them will give a completely wrong answer even if the rest of the working is perfect.

一个常见的陷阱是混淆 x 和 y 的角色。再次检查：如果 x 是椅子，y 是桌子，那么 40x + 30y 表示每把椅子 £40，每张桌子 £30。如果把它们颠倒，即使其余步骤完美，也会得出完全错误的答案。

6. Using the Corner-Point Method | 使用顶点法

At IGCSE level, the optimal solution will always occur at a vertex (corner point) of the feasible region, provided the region is closed and the objective function has a maximum or minimum. The corner-point method involves listing all vertices of the feasible region, substituting each into the objective function, and picking the one that gives the highest or lowest value as required.

在 IGCSE 程度，最优解总是出现在可行域的顶点（角点）处，前提是区域封闭并且目标函数有最大值或最小值。顶点法包括列出可行域的所有顶点，将每个顶点代入目标函数，然后根据要求选取给出最高值或最低值的那个顶点。

For example, if the feasible region has vertices A(0,0), B(0,30), C(20,20), and D(30,0), and the objective is P = 40x + 30y, you would calculate:
At A: P = 0
At B: P = 40×0 + 30×30 = 900
At C: P = 40×20 + 30×20 = 800 + 600 = 1400
At D: P = 40×30 + 0 = 1200
The maximum profit of 1400 occurs at C(20,20).

例如，如果可行域的顶点为 A(0,0), B(0,30), C(20,20) 和 D(30,0)，目标为 P = 40x + 30y，你将计算：
在 A: P = 0
在 B: P = 40×0 + 30×30 = 900
在 C: P = 40×20 + 30×20 = 800 + 600 = 1400
在 D: P = 40×30 + 0 = 1200
最大利润 1400 出现在 C(20,20)。

Always show your substitution clearly, as method marks are awarded for using the correct vertex values. If two vertices give the same optimal value, the entire edge between them will also be optimal, and you may be asked to state that any point along that line segment yields the optimum.

始终清晰地展示代入过程，因为使用正确顶点值可获得方法分。如果两个顶点给出相同的最优值，则它们之间的整条边也将是最优的，你可能需要说明该线段上的任意点均可得出最优值。

7. Dealing with Integer Solutions | 处理整数解

In many contextual problems, the variables x and y must be integers – you cannot manufacture 2.7 chairs or hire 3.5 workers. If the optimal vertex has non-integer coordinates, you must search for the best whole-number point still inside (or on the boundary of) the feasible region.

在许多实际情境中，变量 x 和 y 必须是整数——你不可能生产 2.7 把椅子或雇用 3.5 个工人。如果最优顶点的坐标不是整数，你必须在可行域内（或边界上）寻找最好的整数点。

This is done by listing integer points near the optimal vertex that lie within the region. Test each in the objective function and select the one that gives the best value. For example, if the theoretical maximum is at (15.3, 12.8), test integer points like (15,12), (15,13), (16,12), etc., but ensure they satisfy all inequalities. The integer optimum may not be the nearest grid point – always verify feasibility.

这可以通过列出最优顶点附近且位于区域内的整数点来完成。将每个点代入目标函数，选出给出最佳值的点。例如，如果理论最大值在 (15.3, 12.8)，测试整数点如 (15,12), (15,13), (16,12) 等，但要确保它们满足所有不等式。整数最优点可能不是最邻近的网格点——务必验证可行性。

Sometimes you will be explicitly asked for an integer solution in the question. Even if not asked, consider whether the context forces integer values. WJEC often includes a final part requiring integer adjustments to test your understanding of real-world applicability.

有时题目会明确要求整数解。即使没有要求，也要考虑背景是否迫使值为整数。WJEC 经常在最后一小问要求进行整数调整，以考查你对实际应用的理解。

8. Avoiding Common Pitfalls | 避免常见错误

One of the most common mistakes is misreading the inequality direction. Words like “at least”, “no more than”, “minimum”, and “maximum” must be translated precisely. Create a personal glossary: “at least” means ≥, “at most” means ≤, “more than” means >, “less than” means <.

最常见的错误之一是误读不等式的方向。像“至少”、“不超过”、“最少”、“最多”这样的词必须准确翻译。创建一个自己的术语表：“至少”表示 ≥，“至多”表示 ≤，“超过”表示 >，“少于”表示 <。

Another error is forgetting to plot x ≥ 0 and y ≥ 0, which can lead to the feasible region extending into negative quadrants incorrectly. Also, shading the wrong side of a line can destroy the entire solution. Always test with a point, and consider double-checking your shading with a test coordinate shared across all inequalities.

另一个错误是忘记绘制 x ≥ 0 和 y ≥ 0，这可能导致可行域错误地延伸到负象限。此外，在线条的错误一侧涂阴影会毁掉整个解。始终用测试点进行测试，并考虑使用一个共享测试坐标复查所有不等式的阴影区域。

Scale and accuracy are crucial. If your graph is too small or axes are not evenly scaled, read-offs for vertices become unreliable. Use graph paper if provided, and label the scale clearly. Also, do not confuse the objective function with a constraint – it is not plotted; it is only evaluated at vertices.

刻度和精确度至关重要。如果你的图形太小或坐标轴刻度不均匀，顶点的读取就会不可靠。如果提供坐标纸，务必使用，并清晰标注刻度。同时，不要将目标函数与约束条件混淆——目标函数不需要绘制出来；它仅在顶点处求值。

9. Step-by-Step Problem Strategy | 分步解题策略

Adopting a systematic approach ensures you secure every available mark. Follow these steps for any WJEC linear programming question:

采用系统化的方法可以确保你拿到每一分。对于任何 WJEC 线性规划问题，请遵循以下步骤：

Step 1: Define the decision variables explicitly, stating what x and y represent and their units. Step 2: List all constraints as inequalities, including non-negativity. Step 3: Graph each boundary line accurately using solid/dashed lines as appropriate, and shade the unwanted region. Step 4: Identify and clearly label the feasible region. Step 5: Determine the coordinates of all vertices of the feasible region (solving simultaneous equations where necessary). Step 6: Write down the objective function. Step 7: Substitute each vertex into the objective function and record the value. Step 8: State the optimal solution clearly, including the value of the objective function and the corresponding x and y.

步骤 1：明确定义决策变量，说明 x 和 y 代表什么及其单位。步骤 2：将所有约束条件列为不等式，包括非负条件。步骤 3：准确绘制每条边界线，适当使用实线/虚线，并对不符合区域涂阴影。步骤 4：识别并清晰标注可行域。步骤 5：确定可行域所有顶点的坐标（必要时通过联立方程求解）。步骤 6：写下目标函数。步骤 7：将每个顶点代入目标函数并记录数值。步骤 8：清楚地陈述最优解，包括目标函数的值以及对应的 x 和 y。

Practise this strategy on past paper questions until it becomes automatic. Many top-scoring students create a checklist on the exam paper itself and tick off each step as they go. This reduces anxiety and prevents careless omissions.

在历年真题上练习这一策略，直到它成为习惯。许多高分考生会在试卷上创建一个检查清单，逐项打勾。这可以减轻焦虑并防止粗心的遗漏。

10. Top Exam Tips for WJEC | WJEC 考试高分技巧

WJEC examiners expect clarity and precision. Always use a ruler for all straight lines and a sharp pencil for shading. If you make a mistake with shading, clearly label which region is the feasible one – annotations can save marks if the shading is ambiguous.

WJEC 考官期望清晰和精确。所有直线务必使用直尺绘制，涂阴影使用削尖的铅笔。如果你在涂阴影时出错，清晰地标注哪个是可行域——如果阴影模糊不清，注释可以挽救分数。

Read the final question part carefully: it might ask for “the maximum profit” or “the number of chairs and tables that maximise profit”. Answer precisely – if asked for the number of items, you must state x = … and y = …, not just the maximum profit value.

仔细阅读最后小问：可能会要你求“最大利润”或“使利润最大的椅子数和桌子数”。要精准作答——如果问的是物品数量，你必须陈述 x = … 和 y = …，而不仅仅是最大利润值。

Time management is key. A full linear programming question can be worth 10–15 marks and may take around 15–20 minutes. Do not rush the graph; an inaccurate vertex can cost multiple marks. If you run out of time, at least define variables and write all inequalities – you will pick up method marks.

时间管理是关键。一道完整的线性规划题可能值 10–15 分，用时约 15–20 分钟。不要仓促画图；一个不准确的顶点可能导致多分丢失。如果时间不够，至少定义变量并写出所有不等式——你将获得方法分。

Finally, stay calm and logical. Linear programming is a highly structured topic. Once you master the routine, it becomes one of the most predictable and rewarding questions on the paper.

最后，保持冷静和逻辑。线性规划是一个高度结构化的主题。一旦你掌握了常规步骤，它就会成为试卷上最可预测且最容易拿分的题目之一。

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IGCSE WJEC Maths: Linear Programming – Essential Revision | 线性规划 考点精讲