📚 Linear Programming Key Points | 线性规划考点精讲
Linear programming is a powerful tool in IGCSE Edexcel Mathematics used to find the best possible outcome – such as maximum profit or minimum cost – within a set of constraints represented by linear inequalities. This revision guide breaks down every essential skill you need, from plotting inequalities to interpreting the optimal solution in real-world contexts.
线性规划是IGCSE Edexcel数学中一种强大的工具，用于在由线性不等式表示的约束条件下寻找最佳结果，例如最大利润或最小成本。这篇复习指南分解了你需要掌握的每一个关键技能，从绘制不等式到在实际情境中解读最优解。
1. Understanding Linear Inequalities | 理解线性不等式
A linear inequality in two variables looks like 2x + 3y ≤ 12 or y > x − 1. The solution is a region on the coordinate plane, not just a line. You must always consider which side of the boundary line satisfies the inequality.
一个双变量线性不等式形如 2x + 3y ≤ 12 或 y > x − 1。它的解是坐标平面上的一个区域，而不仅仅是一条线。你必须始终考虑边界线的哪一侧满足不等式。
The boundary line is solid for ≤ or ≥ and dashed for < or >. This small detail is often tested in IGCSE marking schemes.
对于 ≤ 或 ≥，边界线为实线；对于 < 或 >，则为虚线。这个小细节经常在IGCSE评分标准中被考查。
2. Drawing Boundary Lines Accurately | 准确绘制边界线
To draw 4x + 5y = 20, find the intercepts: set x = 0 gives y = 4; set y = 0 gives x = 5. Plot (0,4) and (5,0) and join them with a straight line. For a dashed line, indicate it with small breaks or label it clearly.
要绘制 4x + 5y = 20，找出截距：令 x = 0 得 y = 4；令 y = 0 得 x = 5。标出 (0,4) 和 (5,0)，并用直线连接它们。对于虚线，用小间隔表示或清楚地标注。
Always use a ruler and label the line with its equation. In an exam, neatness saves time when you shade the feasible region later.
始终使用直尺，并用方程标注直线。在考试中，整洁的作图能为你之后给可行域涂色节省时间。
3. Shading the Correct Region | 正确着色区域
After drawing the line, pick a test point – usually (0,0) if it is not on the line. Substitute it into the inequality. If the inequality holds true, shade the side containing the test point; if false, shade the opposite side.
画出直线后，选取一个测试点——如果 (0,0) 不在线上通常使用它。将其代入不等式。如果不等式成立，则着色包含测试点的一侧；如果不成立，则着色另一侧。
For a system of inequalities, you will shade several regions and the feasible region is where all shaded areas overlap. Use distinct shading directions or a single clear colour for the final region.
对于一组不等式，你需要为多个区域着色，而可行域是所有着色区域重叠的部分。使用不同的着色方向，或为最终区域使用一种清晰的单一颜色。
4. Defining the Feasible Region | 定义可行域
The feasible region is the set of all points that satisfy every constraint simultaneously. It is often a polygon bounded by the intersection of boundary lines. IGCSE problems may ask you to mark this region with a capital letter R.
可行域是同时满足所有约束条件的点的集合。它通常是一个由边界线相交围成的多边形。IGCSE题目可能要求你用大写字母 R 标出这个区域。
A feasible region can be unbounded, but in most exam questions it is bounded by axes and inequalities like x ≥ 0, y ≥ 0, making it a closed shape in the first quadrant.
可行域可能是无界的，但在大多数考题中，它被坐标轴以及 x ≥ 0、y ≥ 0 这样的不等式所限制，从而在第一象限形成一个封闭的形状。
5. Introducing the Objective Function | 引入目标函数
The objective function is the expression you aim to maximise or minimise, such as Profit P = 5x + 7y. In IGCSE, it is usually given in the form P = ax + by or C = ax + by.
目标函数是你要最大化或最小化的表达式，例如 利润 P = 5x + 7y。在IGCSE中，它通常以 P = ax + by 或 C = ax + by 的形式给出。
Each point in the feasible region gives a value of the objective function. Your task is to find the point that gives the highest or lowest possible value, depending on the problem’s aim.
可行域中的每一个点都会给出目标函数的一个值。你的任务是找到使目标函数达到最大或最小可能值的那个点，这取决于问题的目标。
6. The Corner Point Method | 顶点法
The maximum or minimum of a linear objective function over a closed feasible region always occurs at a vertex (corner) of the region. This is the cornerstone of linear programming in IGCSE.
线性目标函数在一个闭合可行域上的最大值或最小值总是出现在该区域的某个顶点（角点）上。这是IGCSE线性规划的基石。
To apply the method, find the coordinates of all vertices by solving pairs of simultaneous linear equations that form the boundaries. Then substitute each vertex into the objective function.
要应用该方法，通过解构成边界的二元一次方程组，找出所有顶点的坐标。然后将每个顶点代入目标函数。
| Vertex | x | y | P = 3x + 2y |
| A | 0 | 0 | 0 |
| B | 0 | 4 | 8 |
| C | 3 | 2 | 13 |
| D | 5 | 0 | 15 |
Compare the calculated values – the largest is the maximum, the smallest is the minimum. This method is both exam-friendly and foolproof.
比较计算出的值——最大者即为最大值，最小者即为最小值。这种方法既适合考试，又万无一失。
7. Solving Simultaneous Equations for Vertices | 解方程组求顶点坐标
A vertex is the intersection of two lines. For lines 2x + y = 8 and x + 2y = 10, solve simultaneously. Multiply the second equation by 2: 2x + 4y = 20. Subtract the first: 3y = 12, so y = 4. Substitute back: 2x + 4 = 8, giving x = 2. Vertex is (2,4).
顶点是两条直线的交点。对于直线 2x + y = 8 和 x + 2y = 10，联立求解。将第二个方程乘以2：2x + 4y = 20。减去第一个方程：3y = 12，故 y = 4。代回：2x + 4 = 8，得 x = 2。顶点为 (2,4)。
Always check that the vertex lies within the feasible region; sometimes an intersection point lies outside the region due to another constraint cutting it off – such a point is not a valid corner.
始终检查顶点是否位于可行域内；有时由于另一个约束条件的切割，交点会落在可行域之外——这样的点不是有效的角点。
8. Integer Solutions and Discrete Constraints | 整数解与离散约束
Many real-world problems require integer solutions because you cannot produce half a bicycle or hire 2.3 workers. When the optimal vertex gives a non-integer answer, you must search the integer points near that vertex.
许多实际问题要求整数解，因为你不能生产半辆自行车或雇佣2.3个工人。当最优顶点给出非整数答案时，你必须在该顶点附近搜索整数点。
Draw a small grid around the optimal vertex and test all integer points that are still within the feasible region. Remember: rounding the coordinates of the optimal vertex may give a point outside the region.
在最优顶点周围画一个小网格，并测试所有仍在可行域内的整数点。记住：将最优顶点的坐标四舍五入可能会得到一个位于区域外的点。
Example: If vertex (3.7, 2.2) gives max profit, test (3,2), (3,3), (4,2), (4,3) etc., ensuring each satisfies all inequalities.
示例：如果顶点 (3.7, 2.2) 给出最大利润，测试点 (3,2)、(3,3)、(4,2)、(4,3) 等，确保每个点都满足所有不等式。
9. Interpreting the Optimal Solution | 解读最优解
Once you find the values of x and y that optimise the objective function, answer the original question in words. For a profit problem, state: ‘The maximum profit of $… is achieved by producing … units of X and … units of Y.’
一旦你找到了使目标函数最优化的 x 和 y 值，用文字回答原问题。对于利润问题，表述为：“最大利润为…美元，通过生产…件X和…件Y实现。”
If the problem asks for the maximum value, make sure you give the value of the objective function, not just the coordinates. Lost marks often come from forgetting this final step.
如果问题要求求出最大值，请确保你给出目标函数的值，而不仅仅是坐标。丢分常常是因为忘记了这最后一步。
10. Writing Constraints from Word Problems | 根据应用题写出约束条件
A typical IGCSE question describes resources: time on machine A, machine B, material limit, minimum production requirements. Translate each sentence into an inequality. For instance, ‘each chair requires 2 hours of labour and each table requires 3 hours; total labour available is 60 hours’ becomes 2x + 3y ≤ 60.
典型的IGCSE题目会描述资源：机器A的时间、机器B的时间、材料限制、最低生产要求。将每一句话转化为一个不等式。例如，“每把椅子需要2小时人工，每张桌子需要3小时人工；可用人工总计60小时”转化为 2x + 3y ≤ 60。
Define variables clearly at the start: ‘Let x be the number of chairs and y be the number of tables.’ This makes the modelling step straightforward and earns method marks.
在起始处清楚地定义变量：“设x为椅子数量，y为桌子数量。”这使建模步骤直截了当，并能赢得方法分。
11. Using a Ruler and Spotting Parallel Lines | 使用直尺与识别平行线
When you need to find the optimal point using a sliding ruler method, rewrite the objective function as y = −(a/b)x + P/b. For P = 3x + 2y, this gives y = −(3/2)x + P/2. The gradient is −3/2.
当你需要用推移直尺法寻找最优点时，将目标函数改写为 y = −(a/b)x + P/b。对于 P = 3x + 2y，得到 y = −(3/2)x + P/2。斜率为 −3/2。
Place a ruler at this gradient and slide it parallel across the feasible region. The last point the ruler touches when maximising (or first point when minimising) is the optimal vertex. This geometrical approach is fast and visually confirms your calculations.
将一把直尺按此斜率放置，并在可行域上平行推移。当最大化时，直尺最后接触的那个点（或最小化时最初接触的点）就是最优顶点。这种几何方法快速，并能直观地验证你的计算。
12. Common Pitfalls and Exam Tips | 常见错误与考试技巧
- Always label axes, lines and the feasible region. Unlabelled diagrams lose communication marks.
- Do not shade everything too dark; you must clearly show the feasible region.
- Check that vertices lie inside the feasible region before computing the objective value.
- For integer solutions, do not round the optimal coordinates; test nearby integer points.
- When reading a word problem, highlight numbers and the related quantities to set up inequalities correctly.
- 始终为坐标轴、直线和可行域添加标签。未标注的图会丢失交流沟通分。
- 不要把所有东西都涂得太深；你必须清晰地显示出可行域。
- 在计算目标函数值之前，检查顶点是否位于可行域内。
- 对于整数解，不要对最优坐标进行四舍五入；测试附近的整数点。
- 在阅读应用题时，高亮数字及相关数量，以正确建立不等式。
Practice with past paper questions, especially those that combine setting up constraints and finding optimal integer solutions. These are very common in Edexcel IGCSE 4MA1 examinations.
通过历年真题进行练习，特别是那些结合了建立约束条件与寻找最优整数解的题目。这类题目在Edexcel IGCSE 4MA1考试中非常常见。
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