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

📚 Mastering Linear Programming for GCSE Edexcel Maths | GCSE Edexcel 数学：线性规划考点精讲

Linear programming is a powerful technique used to find the best possible outcome, such as maximum profit or minimum cost, given a set of constraints that can be expressed as linear inequalities. Mastering this topic for your GCSE Edexcel Maths exam means being able to identify variables, write down the constraints, sketch the feasible region, and use either the vertex method or the parallel line method to locate the optimal solution.

线性规划是一种强大的方法，用于在一组线性不等式约束下，寻找最佳结果，比如最大利润或最小成本。要掌握 GCSE Edexcel 数学中这一考点，你需要学会定义变量、列出约束条件、绘制可行域，并运用顶点法或平行线法找到最优解。

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

Linear programming involves maximising or minimising a linear objective function, such as Profit = 3x + 2y, subject to constraints that are linear inequalities like x + y ≤ 10 or 2x + 3y ≤ 24. The variables x and y usually represent quantities of products, hours, or other discrete units, and they are always non‑negative (x ≥ 0, y ≥ 0).

线性规划是对一个线性目标函数（如利润 = 3x + 2y）求最大值或最小值，同时要满足线性不等式约束，例如 x + y ≤ 10 或 2x + 3y ≤ 24。变量 x 和 y 通常代表产品数量、小时数等离散单位，并且总是非负的（x ≥ 0，y ≥ 0）。

In the GCSE Edexcel exam, questions are typically set in a real‑life context: a factory making two types of goods, a farmer planting crops, or a company deciding how many of two services to offer. You will be asked to find the combination that gives the highest profit or the lowest cost, and often you must explain why a given combination is optimal.

在 GCSE Edexcel 考试中，题目通常给出一个真实情境：工厂生产两种产品、农场种植作物、或公司决定提供两种服务的数量。你需要找出带来最高利润或最低成本的组合，并且常常需要解释为什么该组合是最优的。

2. Defining Variables and Constraints | 定义变量与约束条件

Always begin by defining your variables clearly. For example, let x be the number of chocolate cakes and y be the number of vanilla cupcakes produced. Next, translate the verbal restrictions into linear inequalities. Every constraint must be in the form ax + by ≤ c or ax + by ≥ c, using the appropriate inequality sign.

始终要先清晰地定义变量。例如，设 x 为巧克力蛋糕的数量，y 为香草杯子蛋糕的数量。接着，将文字描述的限制条件转化为线性不等式。每个约束都必须写成 ax + by ≤ c 或 ax + by ≥ c 的形式，并且使用正确的不等号。

Common constraints include limits on raw materials, machine time, or labour hours. For instance, if each chocolate cake uses 200 g of flour and each vanilla cupcake uses 100 g, and you have 2000 g of flour available, the flour constraint is 200x + 100y ≤ 2000, which simplifies to 2x + y ≤ 20. Don’t forget the non‑negativity restrictions x ≥ 0 and y ≥ 0, which are always required unless specified otherwise.

常见的约束包括原料、机器时间或人工时数的限制。例如，每个巧克力蛋糕用 200 g 面粉，每个香草杯子蛋糕用 100 g，而你只有 2000 g 面粉，那么面粉约束为 200x + 100y ≤ 2000，化简后得 2x + y ≤ 20。不要忘记非负限制 x ≥ 0 和 y ≥ 0，除非题目另有说明，否则总是需要写上。

3. Drawing Inequality Regions | 绘制不等式区域

To sketch the feasible region for a linear inequality, start by drawing the boundary line as if it were an equation (e.g. 2x + y = 20). Use a solid line if the inequality includes ‘equal to’ (≤ or ≥) and a dashed line if it is a strict inequality (< or >). Then choose a test point not on the line—usually (0,0) is easiest—and check whether it satisfies the inequality. If it does, shade the side containing the test point; if not, shade the other side.

绘制一个线性不等式的可行区域时，要先把边界线当作方程来画（例如 2x + y = 20）。如果不等号包含等于（≤ 或 ≥），用实线；若是严格不等号（< 或 >），用虚线。然后选取不在直线上的一点——通常用 (0,0) 最简便——检验它是否满足不等式。若满足，将包含该点的一侧涂上阴影；若不满足，则涂另一侧。

When several inequalities are plotted on the same grid, the feasible region is the area where all shaded parts overlap. Label this region R often. Ensure your axes are scaled properly and labelled clearly. In the exam, use a sharp pencil and a ruler for neatness, as accurate graphs are essential for obtaining correct solutions.

当多个不等式画在同一坐标平面上时，可行域就是所有阴影重叠的部分，通常标为 R。一定要合理设置坐标轴刻度并清晰标注。考试中用削尖的铅笔和直尺作图，整洁准确的图像是得到正确解答的关键。

4. Identifying the Feasible Region | 确定可行域

The feasible region is a polygon formed by the intersection of all constraints. It represents all possible combinations of x and y that satisfy every restriction simultaneously. In linear programming problems for GCSE, this region is almost always a convex polygon, often a quadrilateral or triangle.

可行域是由所有约束相交形成的多边形，它表示同时满足所有限制的所有 (x, y) 组合。在 GCSE 的线性规划题中，这个区域几乎总是凸多边形，通常是四边形或三角形。

Sometimes the feasible region is unbounded in one direction, but exam questions typically provide enough constraints to form a closed, bounded region. Carefully check the overlap of shading, and label the vertices (corner points) of the feasible region. These vertices are crucial because the optimal value of the objective function will occur at one of them, provided the region is bounded and the objective function is linear.

有时可行域可能朝某一方向无界，但考试题通常会给足约束来形成一个封闭的有界区域。请仔细检查阴影的重叠部分，并将可行域的顶点（角点）标出来。这些顶点非常关键，因为当区域有界且目标函数为线性时，最优值必定出现在某一个顶点上。

5. The Objective Function | 目标函数

The objective function is the expression you want to maximise or minimise, usually given in terms of x and y. For example, if a chocolate cake gives a profit of £5 and a vanilla cupcake gives £2, the profit function is P = 5x + 2y. The goal may be ‘maximise P’ or ‘minimise C’, where C = 3x + 4y represents cost.

目标函数是你希望求最大值或最小值的表达式，通常用 x 和 y 表示。例如，若一个巧克力蛋糕盈利 £5，一个香草杯子蛋糕盈利 £2，利润函数就是 P = 5x + 2y。目标可能是“最大化 P” 或 “最小化 C”，其中 C = 3x + 4y 表示成本。

In GCSE Edexcel questions, the objective function is always linear and is stated directly or can be formed from the given data. Always double‑check whether the question asks for a maximum or a minimum, and make sure the units are consistent. The objective function connects your graph work to the final answer, so take care when writing it down.

在 GCSE Edexcel 试题中，目标函数总是线性的，并直接给出或可根据所给数据建立。务必反复确认题目是求最大值还是最小值，并保证单位一致。目标函数把你画的图形和最终答案连接起来，所以写下来时要格外仔细。

6. The Vertex Method for Optimal Solutions | 顶点法求最优解

The vertex method relies on the corner‑point principle: for a linear objective function over a bounded feasible region, the maximum or minimum will always occur at one of the vertices of the region. Therefore, you simply need to identify all vertices, calculate the value of the objective function at each, and select the largest (for maximisation) or smallest (for minimisation).

顶点法基于角点原理：在有界可行域上，线性目标函数的最大值或最小值总是出现在区域的某个顶点上。因此，你只需找出所有顶点，分别计算每个顶点处的目标函数值，然后选择最大值（对最大化问题）或最小值（对最小化问题）。

To find the coordinates of a vertex where two constraint lines intersect, solve the two equations simultaneously. For example, if one constraint is x + y = 10 and another is 2x + 3y = 24, solve them to get the exact (x, y). Sometimes vertices lie on axes—these are easier: (0,0), (a,0), (0,b). List all vertices with their coordinates, compute the objective function value, and state clearly which vertex gives the optimal solution.

要求出两条约束线相交的顶点坐标，需要联立两个方程求解。例如，一个约束为 x + y = 10，另一个为 2x + 3y = 24，解它们就得到精确的 (x, y)。有时顶点在坐标轴上——这就更容易：(0,0)、(a,0)、(0,b)。列出所有顶点及其坐标，计算目标函数值，并清楚地说明哪个顶点给出最优解。

The vertex method is robust and exam‑friendly. Just be mindful of any additional integer restrictions, which we will discuss later, because the optimal integer solution might not be exactly at a vertex.

顶点法稳健且适合考试。只要留意后面将讨论的整数限制，因为最优整数解不一定正好在顶点上。

7. The Parallel Line Method | 平行线法

As an alternative, the parallel line method involves drawing a line representing the objective function for a convenient constant value, such as P = 0 or P = 10. Then slide this line parallel to itself in the direction that increases (or decreases) the objective function until it just touches the feasible region at the last possible point before leaving it. That point of contact is the optimal solution.

另一种方法是平行线法，即先画一条表示目标函数在某方便常数值下的直线，例如 P = 0 或 P = 10。然后将这条线沿着使目标函数增加（或减少）的方向平行移动，直到它恰好碰到可行域并且在离开区域前的最后一点。那个接触点就是最优解。

To use this method, first rewrite the objective function in the form y = mx + c to see its slope. Draw a line with that slope and label it ‘profit line’ or ‘cost line’. Then move your ruler parallel to this line, observing which vertex is the last to be touched. For maximisation, push the line as far as possible in the direction of increasing P; for minimisation, push it in the direction of decreasing C.

使用此方法时，先将目标函数改写为 y = mx + c 的形式，看清它的斜率。画出具有该斜率的一条直线，标为“利润线”或“成本线”。然后用直尺平行移动这条线，观察最后碰到的是哪个顶点。对于最大化问题，将直线朝使 P 增大的方向推到最远；对于最小化，则朝使 C 减小的方向推动。

This method gives a visual check for the vertex method and can be particularly helpful when you need to explain why a point is optimal without evaluating all vertices. However, both methods lead to the same answer when done correctly.

这种方法可以直观地验证顶点法，当需要解释为什么某点是最优而又不计算所有顶点时尤其有用。但只要操作正确，两种方法会得出相同的答案。

8. Dealing with Integer Constraints | 处理整数约束

In many GCSE problems, x and y represent whole items like numbers of people, cakes, or tables, so they must be integers. The feasible region gives all real‑valued points, but you may need to pick integer coordinates from within or on the boundary of the region. The optimal integer solution is often close to the continuous optimal vertex, but not always exactly that vertex.

在许多 GCSE 问题中，x 和 y 代表整件物品，如人数、蛋糕数或桌子数，因此它们必须是整数。可行域包含所有实数值点，但你可能要从区域内部或边界上选取整数坐标。最优的整数解通常接近连续的顶点最优解，但并不一定正好是该顶点。

If the vertex giving the maximum is, say, (3.6, 7.2), you cannot produce 3.6 cakes. You then need to test integer points around that vertex—typically (3,7), (4,7), (3,8), (4,8)—and see which one still lies in the feasible region and gives the highest value of the objective function. Do not assume that rounding the non‑integer coordinates to the nearest whole number gives the optimum; always check neighbouring integer points.

如果给出最大值的顶点是 (3.6, 7.2)，你不能生产 3.6 个蛋糕。此时需要在该顶点周围测试整数点——通常是 (3,7)、(4,7)、(3,8)、(4,8)——看哪些点仍在可行域内，并给出目标函数的最大值。不要以为把非整数坐标四舍五入到最近的整数就是最优解，一定要检查相邻的整数点。

The exam mark scheme often awards method marks for logically testing integer points, so show your working clearly. Draw a small table listing the candidate integer points and their objective values, and cross out any not meeting constraints.

考试的评分方案通常会给有逻辑地检验整数点的方法分数，所以要清晰地展示步骤。列一个小表格，列出候选整数点及其目标值，并把不满足约束的点划掉。

9. Worked Example: Cake Factory | 实例讲解：蛋糕工厂

A bakery produces chocolate cakes (x) and fruit cakes (y). Each chocolate cake needs 0.5 kg sugar and 0.2 kg butter; each fruit cake needs 0.3 kg sugar and 0.4 kg butter. There are 12 kg of sugar and 8 kg of butter available. The profit on a chocolate cake is £4, and on a fruit cake is £3. The bakery wants to maximise total profit. Write the constraints and find the optimal integer solution.

一家糕点店生产巧克力蛋糕 (x) 和水果蛋糕 (y)。每个巧克力蛋糕需 0.5 kg 糖和 0.2 kg 黄油；每个水果蛋糕需 0.3 kg 糖和 0.4 kg 黄油。现有 12 kg 糖和 8 kg 黄油。巧克力蛋糕的利润为 £4，水果蛋糕为 £3。糕点店希望总利润最大。写出约束条件，并求出最优整数解。

First, define x ≥ 0, y ≥ 0. Sugar constraint: 0.5x + 0.3y ≤ 12 → multiply by 10: 5x + 3y ≤ 120. Butter constraint: 0.2x + 0.4y ≤ 8 → multiply by 10: 2x + 4y ≤ 80 → simplify: x + 2y ≤ 40. Objective: maximise P = 4x + 3y.

首先，定义 x ≥ 0，y ≥ 0。糖约束：0.5x + 0.3y ≤ 12 → 乘 10：5x + 3y ≤ 120。黄油约束：0.2x + 0.4y ≤ 8 → 乘 10：2x + 4y ≤ 80 → 化简：x + 2y ≤ 40。目标函数：最大化 P = 4x + 3y。

Drawing the lines and shading gives a feasible polygon with vertices at (0,0), (24,0), (0,20), and the intersection of 5x + 3y = 120 and x + 2y = 40. Solve simultaneously: from x + 2y = 40, x = 40 – 2y. Substitute into 5x + 3y = 120: 5(40 – 2y) + 3y = 120 → 200 – 10y + 3y = 120 → 200 – 7y = 120 → 7y = 80 → y = 80/7 ≈ 11.43, x = 40 – 2(80/7) = 40 – 160/7 = 120/7 ≈ 17.14. The vertex is (120/7, 80/7).

画出直线并涂阴影后，可行多边形顶点为 (0,0)、(24,0)、(0,20) 以及 5x + 3y = 120 和 x + 2y = 40 的交点。联立求解：由 x + 2y = 40 得 x = 40 – 2y，代入 5x + 3y = 120：5(40 – 2y) + 3y = 120 → 200 – 10y + 3y = 120 → 200 – 7y = 120 → 7y = 80 → y = 80/7 ≈ 11.43，x = 40 – 2(80/7) = 40 – 160/7 = 120/7 ≈ 17.14。顶点为 (120/7, 80/7)。

Since cakes must be integers, we test integer points around this vertex. The feasible integer points near (17.14, 11.43) include (17,11), (17,12), (18,11). Check which satisfy both constraints. For (17,11): sugar 5(17)+3(11)=85+33=118 ≤120, butter 17+2(11)=17+22=39≤40 → OK. P=4(17)+3(11)=68+33=101. For (17,12): sugar 85+36=121>120 → not allowed. For (18,11): sugar 90+33=123>120 → not allowed. Also check (16,12): sugar 5(16)+3(12)=80+36=116≤120, butter 16+2(12)=16+24=40≤40 → OK. P=4(16)+3(12)=64+36=100. So the maximum profit is £101 with 17 chocolate cakes and 11 fruit cakes.

因为蛋糕必须是整数，我们检验该顶点附近的整数点。靠近 (17.14, 11.43) 的可行整数点有 (17,11)、(17,12)、(18,11)。检查它们是否都满足约束。对 (17,11)：糖 5(17)+3(11)=85+33=118 ≤120，黄油 17+2(11)=17+22=39≤40 → 可行，P=4(17)+3(11)=68+33=101。对 (17,12)：糖 85+36=121>120 → 不可行。对 (18,11)：糖 90+33=123>120 → 不可行。再检查 (16,12)：糖 5(16)+3(12)=80+36=116≤120，黄油 16+2(12)=16+24=40≤40 → 可行，P=4(16)+3(12)=64+36=100。所以最大利润为 £101，生产 17 个巧克力蛋糕和 11 个水果蛋糕。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is forgetting to simplify inequalities before graphing, which makes finding intersection points harder. Always divide through by common factors. Another error is misidentifying the feasible region by shading the wrong side of a line—always test with (0,0) if the line does not pass through the origin.

一个常见错误是绘图前忘记化简不等式，导致求交点困难。始终要先约去公因数。另一个错误是涂错阴影方向，把可行域标反——如果直线不经过原点，永远用 (0,0) 测试。

Students often calculate the value of the objective function only at one vertex and assume it is optimal without checking all vertices. Always list every vertex and calculate its objective value, then compare. When integer solutions are required, many candidates simply round the non‑integer optimal vertex and fail to test surrounding points; this can lose marks.

学生经常只计算一个顶点的目标函数值，并想当然认为它是最优的，而没有检查所有顶点。一定要列出每个顶点并计算目标值，再进行比较。当要求整数解时，很多考生只是将非整数最优顶点四舍五入，而未测试周边点，这样可能失分。

Scale and labelling issues can also cause errors. Use sensible scales that make your feasible region large enough to read accurately. Label each constraint line with its equation, and clearly indicate the feasible region R. Show your working for solving simultaneous equations, as method marks are available even if the final graph is slightly inaccurate.

坐标刻度和标注问题也会引发错误。使用合理的刻度，使可行域足够大，便于精确读取。给每条约束直线标上方程，并清晰标示出可行域 R。展示解联立方程的过程，因为即便最终图形稍有不准，也可得到方法分。

11. Exam Tips for Edexcel GCSE | Edexcel GCSE 考试技巧

Linear programming questions often appear in the higher tier and can carry many marks. Read the question carefully to identify exactly what needs to be maximised or minimised. Underline key numbers and phrases. Always show the definitions ‘Let x = …’ and ‘Let y = …’ at the beginning—this earns marks and organises your thinking.

线性规划题常在高级别试卷中出现，占分较多。仔细读题，准确找出需要最大化或最小化的对象。在关键数字和短语下划线。开头一定要写出定义“设 x = …”和“设 y = …”——这能得分，也有助于理清思路。

When graphing, use graph paper or draw axes carefully. Plot points for each constraint line, but do not waste time drawing every line in full—just the segment that contributes to the feasible region is enough, though drawing the whole line is accepted. After identifying the optimal solution, restate it clearly in the context of the question, complete with units. For example, ‘The maximum profit is £101, achieved by making 17 chocolate cakes and 11 fruit cakes.’

作图时使用坐标纸或细心画出坐标轴。为每条约束线描点，但不必浪费时间画出完整的整条线——只需画出构成可行域边界的线段，不过画整条线也能接受。找到最优解后，要结合题目语境清楚地复述出来，并带上单位。例如，“最大利润为 £101，通过制作 17 个巧克力蛋糕和 11 个水果蛋糕实现。”

Check your final answer against all constraints as a quick verification. This can catch silly arithmetic errors and ensure your solution is truly feasible. A few seconds of checking can prevent lost marks.

快速验证最终答案是否满足所有约束，这能发现愚蠢的计算错误，并确保解是真正可行的。花几秒钟检查就能避免失分。

12. Summary and Key Takeaways | 总结与核心要点

Linear programming is a systematic method for finding the best outcome under given linear constraints. Remember the core steps: define variables, write all inequalities (including non‑negativity), graph the feasible region, find the coordinates of all vertices, evaluate the objective function at each vertex, and choose the optimum. If integer solutions are needed, test nearby integer points.

线性规划是一种在给定线性约束下寻找最佳结果的系统方法。记住核心步骤：定义变量，写出所有不等式（包括非负性），画出可行域，求出所有顶点坐标，在每个顶点计算目标函数值，并选出最优值。如果需要整数解，再检验附近的整数点。

Mastering both the vertex method and the parallel line method gives you flexibility in the exam. Practise constructing inequalities from word problems, as this is a skill that underpins the whole topic. With clear working and careful graphing, you can secure high marks on this rewarding part of the GCSE Edexcel Maths syllabus.

同时掌握顶点法和平行线法，能让你在考试中灵活应对。多练习从文字题建立不等式，这是支撑整个专题的关键技能。有了清晰的步骤和细致的作图，你就能在 GCSE Edexcel 数学大纲这个回报丰厚的部分稳稳拿下高分。

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Mastering Linear Programming for GCSE Edexcel Maths | GCSE Edexcel 数学：线性规划 考点精讲