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

📚 GCSE OCR Maths: Linear Programming | GCSE OCR 数学：线性规划考点精讲

Linear programming is a powerful technique used in GCSE OCR Mathematics to solve real-world problems where you need to maximise or minimise a quantity, such as profit or cost, under certain restrictions. The restrictions are expressed as linear inequalities, and the goal is to find the best possible combination of two variables that satisfies all of them simultaneously. This revision guide will walk you through each step, from setting up constraints to finding the optimal solution, with clear explanations and a worked example.

线性规划是GCSE OCR 数学中一个强有力的工具，用于解决现实世界中需要在特定限制条件下最大化或最小化某个量（如利润或成本）的问题。这些限制条件用线性不等式表示，而目标是找到同时满足所有条件的两个变量的最佳组合。本考点精讲将带你逐步走过每一个步骤，从列出约束条件到找出最优解，并配有清晰的解释和实例解析。

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

Linear programming is a method for finding the optimal value (maximum or minimum) of a linear expression, called the objective function, subject to a set of linear constraints. In GCSE questions, we usually deal with two variables, often x and y, which represent quantities like the number of items to produce or the amount of resources to use.

线性规划是一种在满足一系列线性约束条件的前提下，寻找线性表达式（称为目标函数）最优值（最大值或最小值）的方法。在GCSE题目中，我们通常处理两个变量，通常是 x 和 y，它们表示诸如要生产的产品数量或要使用的资源量之类的量。

The constraints are written as linear inequalities using symbols like ≤, ≥, . The set of points that satisfy all constraints forms a region on a graph, known as the feasible region. Every point in this region represents a possible solution, but the best one will always lie at a corner (vertex) of the region, provided the region is bounded and the variables are continuous.

约束条件用线性不等式表示，使用 ≤、≥、< 或 > 等符号。满足所有约束条件的点集合在图上形成一个区域，称为可行域。该区域中的每一个点都代表一个可能的解，但最优解总是出现在区域的角点（顶点），前提是该区域有界且变量是连续的。

2. Formulating Constraints | 列出约束条件

Carefully read the problem to identify all the restrictions. Typical constraints include non‑negativity (x ≥ 0, y ≥ 0), limited resources (e.g., 2x + y ≤ 20), minimum requirements (e.g., x + y ≥ 10), or ratios between variables. Write each constraint as a linear inequality.

仔细阅读题目，找出所有限制条件。常见的约束条件包括非负性（x ≥ 0, y ≥ 0）、有限资源（例如 2x + y ≤ 20）、最低要求（例如 x + y ≥ 10）或变量之间的比例。将每个约束条件写成一个线性不等式。

Sometimes the problem involves integer-only solutions (e.g., number of people, cars). Although the graphical method works for continuous variables, if the context demands whole numbers, you may need to check the integer points near the optimal vertex. However, most OCR GCSE questions allow fractional solutions unless clearly stated otherwise.

有时问题只涉及整数解（例如人数、车辆数）。尽管图解法适用于连续变量，但如果情境要求整数，你可能需要检查最优顶点附近的整数点。不过，大多数OCR GCSE题目允许分数解，除非明确说明必须为整数。

3. Graphing Linear Inequalities | 绘制线性不等式图

To represent an inequality on a graph, start by drawing the corresponding boundary line as if it were an equation. Use a solid line for ≤ or ≥ and a dashed line for < or >. Next, choose a test point not on the line – the origin (0,0) is usually the easiest – and substitute it into the inequality. If the test point satisfies the inequality, shade the side containing it; otherwise, shade the opposite side.

要在图上表示一个不等式，首先画出相应的边界线，就像它是方程一样。对于 ≤ 或 ≥ 使用实线，对于 < 或 > 使用虚线。然后，选择一个不在直线上的测试点——原点 (0,0) 通常是最简单的——将其代入不等式。如果测试点满足不等式，则给包含该点的一侧涂上阴影；否则，给另一侧涂上阴影。

When drawing several inequalities on the same grid, you can shade the unwanted regions instead of the wanted one. This leaves the feasible region unshaded, which often makes it easier to see the vertices. OCR exam papers frequently use this ‘leaving unshaded’ approach.

当在同一坐标网格上绘制多个不等式时，你可以给不想要的区域涂上阴影，而保留想要的区域。这样可行域就是未涂色的，通常更容易看清顶点。OCR试卷常用这种“不涂可行域”的方法。

4. Identifying the Feasible Region | 确定可行域

The feasible region is the intersection of all the shaded (or unshaded) regions from your constraints. It is the common area where all inequalities hold true. In two variables, this region is typically a convex polygon. Any point inside or on the boundary of this polygon is a valid solution to the constraints.

可行域是所有约束条件的阴影区域（或未涂色区域）的交集。它是所有不等式都成立的公共区域。在两个变量的情况下，这个区域通常是一个凸多边形。该多边形内部或边界上的任何点都是约束条件的有效解。

Label the vertices clearly. These corner points will be essential in the next step because the optimal solution of a linear programming problem, when it exists, is always achieved at a vertex of the feasible region.

清楚标记顶点。这些角点在下一步至关重要，因为线性规划问题的最优解（如果存在）总是在可行域的顶点处取得。

5. The Objective Function | 目标函数

The objective function is the expression you need to optimise — maximise or minimise. It is given in the form P = ax + by or C = ax + by. Replace P or C with the actual quantity, such as profit, cost, or time.

目标函数是你需要优化的表达式——最大化或最小化。它以 P = ax + by 或 C = ax + by 的形式给出。用实际的量替换 P 或 C，例如利润、成本或时间。

For example, if a company makes £5 profit per chair (x) and £3 profit per table (y), the objective function to maximise is:

例如，如果一家公司每把椅子 (x) 获利5英镑，每张桌子 (y) 获利3英镑，那么需要最大化的目标函数是：

Maximise P = 5x + 3y

Note that the objective function does not have an inequality sign; it is an expression whose value changes depending on x and y. We evaluate this expression at the vertices of the feasible region to find the optimal value.

请注意，目标函数没有不等号；它是一个取决于 x 和 y 而变化的表达式。我们在可行域的顶点处计算该表达式，以找到最优值。

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

The most reliable method for GCSE is the vertex method. Once you have drawn the feasible region and labelled all its vertices, calculate the value of the objective function at each vertex. The largest value gives the maximum, and the smallest gives the minimum.

对于GCSE来说，最可靠的方法是顶点法。一旦你画出了可行域并标出了所有顶点，就在每个顶点处计算目标函数的值。最大的值给出最大值，最小的值给出最小值。

To find the coordinates of a vertex that is not on an axis, solve the two equations of the boundary lines that meet at that point simultaneously. For example, if lines L₁: x + y = 6 and L₂: 2x + y = 8 intersect, subtract one equation from the other to find x and then y.

要找到不在坐标轴上的顶点坐标，解出在该点相交的两条边界线的方程组成的方程组。例如，如果直线 L₁: x + y = 6 和 L₂: 2x + y = 8 相交，将两式相减求出 x，然后求出 y。

Remember, you only need to test the vertices. You do not need to test points inside the feasible region, because a linear function on a convex polygon always attains its extreme values at the corners.

请记住，你只需要检验顶点。你不需要检验可行域内部的点，因为凸多边形上的线性函数总是在角点处取得极值。

7. Using the Sliding Line Method (for Maximisation) | 使用滑动直线法（最大化）

An alternative geometric method is the sliding line. Draw the line for the objective function with a chosen value, e.g., 5x + 3y = 15. Then use a ruler to slide this line parallel to itself in the direction that increases P (for maximisation) or decreases P (for minimisation) until it is just about to leave the feasible region. The last point of contact is the optimal vertex.

一种替代的几何方法是滑动直线法。为目标函数选定某个值画一条直线，例如 5x + 3y = 15。然后用直尺平行移动这条直线，朝着使 P 增大（最大化时）或使 P 减小（最小化时）的方向，直到它即将离开可行域。最后接触的点就是最优顶点。

This method is particularly useful for visualising why the vertex gives the optimum, and it works well when the feasible region is clearly drawn. However, in exams, the vertex method is often faster and less prone to inaccuracy from hand‑drawn graphs.

这种方法特别有利于直观理解为什么顶点给出最优值，并且在可行域画得清晰时效果很好。然而，在考试中，顶点法通常更快，并且不容易因手绘图形而产生误差。

8. Interpreting Solutions in Context | 结合实际情境解释解

After finding the optimal x and y, make sure you answer the original question. State clearly how many of each item should be made, the maximum profit, or the minimum cost. Include units where appropriate.

找到最优的 x 和 y 后，确保你回答了原题。清楚说明每种物品应生产多少、最大利润或最小成本是多少。适当时包含单位。

For example: ‘The company should produce 2 chairs and 4 tables to achieve a maximum profit of £22.’ If the problem asks for integer solutions but your optimal vertex gives a fractional answer, check nearby integer points that still lie inside the feasible region.

例如：“公司应生产2把椅子和4张桌子，以获得最大利润22英镑。”如果问题要求整数解，但你的最优顶点给出的答案是分数，则检查附近仍然位于可行域内的整数点。

9. Worked Example | 实例解析

A factory produces two types of products: A (x units) and B (y units). The constraints are:

x ≥ 0, y ≥ 0
x + y ≤ 6 (labour hours)
2x + y ≤ 8 (machine time)

一家工厂生产两种类型的产品：A（x 件）和 B（y 件）。约束条件为：

x ≥ 0, y ≥ 0
x + y ≤ 6（劳动小时）
2x + y ≤ 8（机器时间）

The profit from product A is £5 per unit, and from product B is £3 per unit. The objective function is to maximise:

产品 A 的利润为每件5英镑，产品 B 的利润为每件3英镑。需要最大化的目标函数为：

P = 5x + 3y

Plot the constraints: Draw x + y = 6 (solid line) and 2x + y = 8 (solid line), along with axes. Shade the unwanted regions. The feasible region is a quadrilateral with vertices at (0,0), (0,6), (2,4), and (4,0). To find the intersection (2,4), solve x + y = 6 and 2x + y = 8 by subtracting: (2x + y) – (x + y) = 8 – 6 → x = 2, then y = 4.

绘制约束条件：画出 x + y = 6（实线）和 2x + y = 8（实线）以及坐标轴。给不想要的区域涂上阴影。可行域是一个四边形，顶点在 (0,0)、(0,6)、(2,4) 和 (4,0)。要求出交点 (2,4)，解方程组 x + y = 6 和 2x + y = 8，相减得 (2x + y) – (x + y) = 8 – 6 → x = 2，然后 y = 4。

Now evaluate P at each vertex:

Vertex (x, y)	P = 5x + 3y
(0,0)	0
(0,6)	18
(2,4)	22
(4,0)	20

The maximum profit is 22 at (2,4). Therefore, the factory should produce 2 units of A and 4 units of B. This interpretation satisfies all constraints and gives the highest possible profit.

最大利润为 22，在 (2,4) 点取得。因此，工厂应生产 2 件 A 和 4 件 B。这种解释满足所有约束条件，并给出尽可能最高的利润。

10. Checking Integer Solutions When Required | 整数解检查（需要时）

In some contexts, x and y must be whole numbers. If the optimum vertex has fractional coordinates, e.g., (2.5, 3.5), you must test integer points around it. List all integer pairs in the feasible region near that vertex, substitute them into the objective function, and select the best one.

在某些情境下，x 和 y 必须是整数。如果最优顶点的坐标是分数，例如 (2.5, 3.5)，你必须检验其周围的整数点。列出该顶点附近可行域内的所有整数对，代入目标函数，并选择最佳的一个。

For the worked example above, the solution was already integral. If the intersection had been (2.5, 3.5), you would check points like (2,3), (2,4), (3,3), etc., ensuring they are inside the region. Sometimes the optimal integer solution is not simply the rounded vertex.

对于上述实例，解已经是整数。如果交点是 (2.5, 3.5)，你将检验诸如 (2,3)、(2,4)、(3,3) 等点，确保它们在区域内。有时最优整数解并非简单地四舍五入顶点坐标。

11. Common Mistakes & How to Avoid Them | 常见错误及如何避免

Mistake 1: Forgetting the non‑negativity constraints x ≥ 0, y ≥ 0. Many problems imply these, but you must include them; otherwise, the feasible region may extend into negative quadrants.

错误1：忘记非负性约束条件 x ≥ 0, y ≥ 0。许多问题暗示了这些条件，但你必须将它们包括在内；否则，可行域可能延伸到负象限。

Mistake 2: Shading the wrong side. Always test a point (like (0,0)) to confirm which side satisfies the inequality. Double-check especially when the inequality is rearranged.

错误2：阴影涂错了侧。始终测试一个点（如 (0,0)）以确认哪一侧满足不等式。当不等式被重新排列时，尤其要仔细检查。

Mistake 3: Confusing solid and dashed lines. Solid line means the boundary is included (≤,≥); dashed line means it is not (). In most practical problems, constraints are inclusive, but be precise.

错误3：混淆实线和虚线。实线表示边界包含在内（≤、≥）；虚线表示边界不包含在内（）。在大多数实际问题中，约束条件是包含边界的，但要准确区分。

Mistake 4: Miscalculating vertex coordinates. Solve the two boundary equations carefully. A common slip is an arithmetic error in simultaneous equations. Always substitute your coordinates back into both equations to verify.

错误4：顶点坐标计算错误。仔细求解两条边界线方程。常见的疏漏是联立方程中的算术错误。始终将坐标代回两个方程进行验证。

Mistake 5: Forgetting to answer the question in context. Even if you find the correct vertex, you must state what the values mean in terms of the original problem.

错误5：忘记结合情境回答问题。即使你找到了正确的顶点，也必须说明这些值在原题背景下的含义。

12. Summary & Exam Tips | 总结与考试技巧

Linear programming in GCSE OCR Maths is all about precise graphical work and systematic evaluation. Practise drawing clean, well‑labelled graphs with a ruler. When revising, work through several past paper questions to become fluent in translating worded constraints into inequalities.

GCSE OCR 数学中的线性规划关键在于精确的绘图和系统的评估。练习用直尺绘制清晰、标注良好的图形。复习时，多做几道历年真题，以便熟练地将文字约束条件转化为不等式。

In the exam, show all your working: draw the lines, label the feasible region, list the vertices, compute the objective function at each vertex, and state your conclusion clearly. This methodical approach will secure full marks even if a small graph slip occurs.

在考试中，展示所有解题步骤：画出直线，标示可行域，列出顶点，在每个顶点处计算目标函数，并清楚地陈述结论。这种有条不紊的方法即使在图形出现小失误时也能确保你拿到满分。

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