Mixed Exercise 5: Straight Line Graphs | 混合练习5：直线图形

📚 Mixed Exercise 5: Straight Line Graphs | 混合练习5：直线图形

Mixed Exercise 5 in the Edexcel AS and A Level Pure Mathematics textbook is the final set of problems for Chapter 5, which focuses on straight line graphs. This exercise is designed to consolidate all the key skills you have developed: calculating gradients, writing equations of lines in various forms, using parallel and perpendicular slopes, finding midpoints and distances, determining intersections, and applying linear models. Mastering the mixed exercise is essential for exam success, as it blends straightforward calculations with problem‑solving and proof tasks that often appear in timed assessments. In this article, we will walk through each concept tested in Mixed Exercise 5, provide clear explanations, and show you how to avoid common pitfalls — all while building the bilingual fluency you need for top marks.

混合练习5是Edexcel AS与A Level纯数学教材第5章“直线图形”末尾的综合练习。它把所有核心技能集中在一起：求斜率、用不同形式写出直线方程、运用平行与垂直的斜率关系、计算中点与距离、求交点以及建立线性模型。要吃透这份练习，不能只记住公式，你必须能把公式灵活地用在需要推理、证明和实际情境的题目里。这篇文章会逐项拆解混合练习5涉及的每一个知识点，给出清晰的英文和中文解说，并指正常见的失误，帮助你在复习中既加深理解又提升应试能力。

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1. What Mixed Exercise 5 Covers | 混合练习5涵盖内容

Mixed Exercise 5 brings together every skill from the chapter on straight line graphs. You will encounter questions that ask you to find the gradient between two points, determine the equation of a line given a point and a gradient or two points, decide whether lines are parallel or perpendicular, calculate the midpoint and length of a segment, and solve linear simultaneous equations to find intersection points. Some questions also ask you to use straight lines to model real‑world situations, such as converting between temperature scales or analysing cost functions. Because the exercise is mixed, you must be ready to switch rapidly between these ideas without being told which method to use — exactly like in an exam paper.

混合练习5将直线图形一章的全部技能整合在一起。题目会要求你计算两点间的斜率、根据给定的点和斜率（或两点）求出直线方程、判断两条直线是否平行或垂直、计算线段的中点与长度，并通过解线性联立方程求交点。部分题目还会让你用直线建立实际模型，比如温度换算或成本分析。因为是综合练习，你必须能够在不同概念之间快速切换，而且题面不会明确告诉你该用哪种方法——这和真实考试完全一致。

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2. Gradient of a Straight Line | 直线的斜率

The gradient measures how steep a line is. Given two points A(x₁, y₁) and B(x₂, y₂), the gradient m is calculated using the formula m = (y₂ − y₁) ÷ (x₂ − x₁). It does not matter which point you call ‘first’, provided you subtract the coordinates in the same order. If the line is horizontal, y₂ − y₁ = 0 and the gradient is 0. If the line is vertical, the denominator x₂ − x₁ = 0, so the gradient is undefined. Mixed Exercise 5 often starts with straightforward gradient calculations, but you may also need to use the gradient to prove that three points are collinear — if the gradients between pairs of points are equal, the points lie on the same line.

斜率衡量直线的倾斜程度。给定两点 A(x₁, y₁) 和 B(x₂, y₂)，斜率 m 用公式 m = (y₂ − y₁) ÷ (x₂ − x₁) 计算。只要保持坐标相减的顺序一致，哪一点作为“第一点”都可以。水平线满足 y₂ − y₁ = 0，斜率为 0；垂直线分母 x₂ − x₁ = 0，斜率无定义。混合练习5经常从简单的斜率计算入手，但你也可能需要利用斜率证明三点共线——如果任意两点间的斜率相等，这些点就在同一条直线上。

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3. Equation of a Straight Line | 直线方程

There are three main forms for the equation of a straight line. The first is y = mx + c, where m is the gradient and c is the y‑intercept. This form is ideal when you can read the intercept directly from a graph or when you are comparing two lines. The second is the point–gradient form y − y₁ = m(x − x₁), which is extremely useful when you know one point on the line and the gradient. Finally, the general form ax + by + c = 0 often appears in coordinate geometry questions, especially when integer coefficients are preferred. In Mixed Exercise 5, you may be asked to write your answer in a specific form, so always read the question carefully.

直线方程有三种常见形式。第一种是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。当你能够直接从图上读出截距，或者需要比较两条直线时，这种形式最方便。第二种是点斜式 y − y₁ = m(x − x₁)，当你已知直线上一点和斜率时，这一形式尤其好用。最后一种是一般式 ax + by + c = 0，常在坐标几何题中出现，特别是当题目要求系数为整数时。在混合练习5中，题目可能会指定最终答案的书写形式，因此一定要仔细审题。

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4. Parallel and Perpendicular Lines | 平行线与垂直线

Two lines are parallel if and only if their gradients are equal: m₁ = m₂. If two distinct lines have the same gradient, they will never intersect. Perpendicular lines have gradients that multiply to give −1: m₁ × m₂ = −1. This means that the gradient of a line perpendicular to a given line is the negative reciprocal, i.e. m₂ = −1/m₁. Mixed Exercise 5 frequently asks you to find the equation of a line that is parallel or perpendicular to a given line and passes through a particular point. You must first identify the gradient of the given line, then apply the parallel or perpendicular condition, and finally use the point–gradient form to write the equation.

两条直线平行当且仅当它们的斜率相等：m₁ = m₂。如果两条不同的直线斜率相同，它们永远不会相交。垂直直线的斜率乘积等于 −1：m₁ × m₂ = −1。这意味着与已知直线垂直的直线，其斜率是已知斜率的负倒数，即 m₂ = −1/m₁。混合练习5经常要求你求一条直线方程，它与给定的直线平行或垂直，且经过某一点。你需要先确定已知直线的斜率，再运用平行或垂直条件，最后用点斜式写出方程。

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5. Midpoint and Distance | 中点与距离

The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is found by averaging the coordinates: Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula is often used to find the centre of a line segment or to work backwards when one endpoint is missing. The distance between the same two points comes from Pythagoras’ theorem: Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]. You will see these formulas applied in Mixed Exercise 5 to decide if a triangle is right‑angled, to find the length of a median, or to check whether a point lies on a circle’s diameter. Always leave lengths in simplified surd form unless the question asks for a decimal approximation.

连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点，可通过坐标取平均得到：中点 = ((x₁ + x₂)/2, (y₁ + y₂)/2)。这个公式常用来求线段的中心，或者在已知一个端点时反向推算。两点间的距离则来自勾股定理：距离 = √[(x₂ − x₁)² + (y₂ − y₁)²]。在混合练习5中，你会用到这些公式来判断三角形是否为直角三角形、求中位线的长度，或者检验某点是否在圆的直径上。除非题目要求小数近似值，否则长度结果应保留为最简根式。

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6. Intersection of Lines | 直线的交点

To find where two straight lines intersect, solve their equations simultaneously. This usually means solving two linear equations in two unknowns, x and y. The substitution method works well when one equation is already in the form y = mx + c, but when both lines are given in general form, elimination is often quicker. If solving leads to a contradiction such as 0 = 5, the lines are parallel and there is no intersection. If the two equations simplify to the same line, there are infinitely many points of intersection. Mixed Exercise 5 may ask for the coordinates of the point where two lines cross, or it may embed intersection tasks within a larger problem about areas of triangles or perpendicular bisectors.

要求两条直线的交点，就要联立解它们的方程。这通常意味着求解两个未知数 x 和 y 的线性方程组。如果其中一个方程已经是 y = mx + c 的形式，代入法非常方便；但如果两条直线都用一般式给出，消元法往往更快。如果求解过程中出现类似 0 = 5 的矛盾结果，说明两线平行，没有交点。如果两个方程化简后表示同一条直线，交点就有无穷多个。混合练习5可能直接要求你写出两条直线交点的坐标，也可能把求交点融入到三角形面积或垂直平分线这样更大的问题里。

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7. Modelling with Straight Lines | 直线模型应用

Many real‑world relationships can be approximated by a straight line. In Mixed Exercise 5, modelling questions might give you two data points — for example, the cost of hiring a bike for different lengths of time — and ask you to find a linear equation connecting the variables. Once you have the equation, you can use it to make predictions. It is important to interpret the gradient and intercept in context: the gradient represents the rate of change (e.g. cost per hour), and the intercept gives the starting value (e.g. fixed hire charge). Always check that your predicted value is within the range of the data, because extrapolating outside that range can be unreliable.

现实世界中许多关系都可以用直线近似表示。混合练习5里的建模题可能会给出两个数据点——比如租用自行车不同时长对应的费用——然后要求你建立连接变量的线性方程。得到方程后，你就可以用它进行预测。务必结合实际情境解释斜率和截距的含义：斜率表示变化率（比如每小时费用），截距表示初始值（比如固定租金）。要特别注意，预测值不应超出已有数据的范围，因为外推可能不准确。

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8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

One frequent mistake in Mixed Exercise 5 is confusing the sign when using the negative reciprocal for perpendicular gradients. If a line has gradient 3, the perpendicular gradient is −1/3, not −3. Another common error is forgetting to check the requested form of the equation — writing y = 2x + 3 when the question demands the form ax + by + c = 0 could cost marks even if the answer is essentially correct. Students also sometimes misapply the midpoint formula by subtracting coordinates instead of adding. Finally, when solving simultaneous equations, always substitute your solution back into both original equations to verify that it works; a simple arithmetic slip can lead you to an intersection point that lies on one line but not the other.

混合练习5中一个常见错误是处理垂直斜率时混淆符号。如果一条直线的斜率为3，与之垂直的斜率是 −1/3，而不是 −3。另一个常见失误是忘记检查方程要求的形式——题目要求写成 ax + by + c = 0 的形式，你却写成 y = 2x + 3，尽管答案本质上正确也会被扣分。还有同学在使用中点公式时误用坐标相减而非相加。最后，在解联立方程时，一定要把解代回两个原方程检验；一个简单的计算错误就可能让你求出一个只在一条直线上而在另一条上不成立的交点。

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9. Worked Example Inspired by Mixed Exercise 5 | 混合练习5典型例题演示

The points P(2, 5) and Q(−4, −3) are given. Find the gradient of the line segment PQ, the equation of the perpendicular bisector of PQ, and the coordinates of the point where this perpendicular bisector crosses the x‑axis. First, calculate the gradient of PQ: m_PQ = (−3 − 5) ÷ (−4 − 2) = −8 ÷ −6 = 4/3. The perpendicular gradient m_perp = −3/4. The midpoint of PQ is ((2 + (−4))/2, (5 + (−3))/2) = (−1, 1). The perpendicular bisector passes through (−1, 1) with gradient −3/4, so its equation is y − 1 = −3/4 (x + 1). Multiply by 4 to avoid fractions: 4y − 4 = −3(x + 1) → 4y − 4 = −3x − 3 → 3x + 4y − 1 = 0. To find the x‑intercept, set y = 0: 3x − 1 = 0 → x = 1/3. The crossing point is (1/3, 0).

已知点 P(2, 5) 和 Q(−4, −3)，求线段 PQ 的斜率、PQ 的垂直平分线方程，以及这条垂直平分线与 x 轴的交点坐标。先求 PQ 斜率：m_PQ = (−3 − 5) ÷ (−4 − 2) = −8 ÷ −6 = 4/3。垂直斜率 m_perp = −3/4。PQ 中点坐标为 ((2 + (−4))/2, (5 + (−3))/2) = (−1, 1)。垂直平分线经过 (−1, 1) 且斜率为 −3/4，因此方程为 y − 1 = −3/4 (x + 1)。两边同乘4消去分母：4y − 4 = −3(x + 1) → 4y − 4 = −3x − 3 → 3x + 4y − 1 = 0。令 y = 0 求 x 轴截距：3x − 1 = 0 → x = 1/3。交点坐标为 (1/3, 0)。

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10. Summary of Key Formulas and Strategy | 关键公式与应考策略

Having a clear overview of the formulas that appear in Mixed Exercise 5 makes revision much more efficient. The table below gathers the essential equations. For the exam, always begin by sketching a quick diagram; label the points, write down what you know, and identify which formula applies. This discipline will prevent careless errors and help you spot whether your final answer makes sense.

清楚梳理混合练习5涉及的公式能极大提高复习效率。下表汇总了最重要的方程式。在考场上，先快速画出示意图，标出各点，写下已知信息，再确定该用哪一个公式。养成这样的习惯可以避免粗心出错，并帮你判断最终答案是否合理。

Concept | 概念	Formula | 公式
Gradient between two points (x₁, y₁) and (x₂, y₂) | 两点间斜率	m = (y₂ − y₁)/(x₂ − x₁)
Equation: slope–intercept form | 斜截式	y = mx + c
Equation: point–gradient form | 点斜式	y − y₁ = m(x − x₁)
Parallel lines condition | 平行条件	m₁ = m₂
Perpendicular lines condition | 垂直条件	m₁ × m₂ = −1 or m₂ = −1/m₁
Midpoint of a segment | 线段中点	M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance between two points | 两点间距离	d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Remember, Mixed Exercise 5 is not just a list of tasks; it is a test of how flexibly you can use these tools. Regular practice, careful reading, and thorough checking will help you secure full marks every time a straight line graph question appears.

请记住，混合练习5不仅仅是一组练习，它是对你能否灵活运用这些工具的检验。坚持练习、仔细审题、全面检查，就能让你在每次遇到直线图形题时稳稳拿到满分。

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