📚 Chapter Review 5: Straight Line Graphs | 第五章复习：直线图
This revision guide covers the key ideas in Edexcel A-Level Pure Mathematics Chapter Review 5: straight line graphs. You will revise gradient, midpoint, distance, equations of lines in several forms, parallel and perpendicular conditions, intersections, linear modelling, sketching, and the exam technique that turns these tools into reliable marks.
本复习指南涵盖 Edexcel A-Level 纯数学第五章复习：直线图。你将复习斜率、中点、距离、直线的多种方程形式、平行与垂直条件、交点、线性建模、画图，以及把这些工具转化为稳定得分的考试技巧。
1. Gradient and Midpoint | 斜率与中点
The gradient of a straight line measures how steeply the line rises or falls. For two points A(x₁, y₁) and B(x₂, y₂), the gradient is the change in y divided by the change in x. You must subtract the coordinates in the same order in the numerator and the denominator; reversing one of them but not the other produces the wrong sign.
直线的斜率衡量直线上升或下降的陡峭程度。对于两点 A(x₁, y₁) 与 B(x₂, y₂)，斜率等于 y 的变化量除以 x 的变化量。分子与分母必须按相同顺序相减；只颠倒其中一个会得到错误的符号。
m = (y₂ − y₁) / (x₂ − x₁)
e order gives the same value, because both numerator and denominator change sign together.
例如，若 A(2, 3)、B(6, 11)，则斜率为 (11 − 3) / (6 − 2) = 8 / 4 = 2。若把顺序完全颠倒，结果相同，因为分子与分母同时变号。
The sign of the gradient tells you the direction of the line. A positive gradient rises from left to right, a negative gradient falls from left to right, a zero gradient gives a horizontal line, and a vertical line has an undefined gradient because the change in x is zero.
斜率的符号说明直线的方向。正斜率从左到右上升，负斜率从左到右下降，斜率为零时直线水平，而垂直直线的斜率无定义，因为 x 的变化量为零。
The midpoint M of a line segment is the point exactly halfway between the two endpoints. Its coordinates are the averages of the corresponding coordinates of the endpoints.
线段的中点 M 位于两个端点正中间，其坐标分别是两个端点对应坐标的平均值。
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
ce that the midpoint lies on the line segment, so substituting it into the equation of the line must give a true statement.
对 A(2, 3) 与 B(6, 11)，中点为 ((2 + 6) / 2, (3 + 11) / 2) = (4, 7)。注意中点在线段上，因此把它代入直线方程应得到恒等式。
2. Distance Between Two Points | 两点间的距离
The distance between two points is found by applying Pythagoras theorem to the horizontal and vertical differences. The horizontal difference is x₂ − x₁ and the vertical difference is y₂ − y₁, so the distance is the square root of the sum of their squares.
两点间的距离可通过把勾股定理应用于水平差与垂直差来求得。水平差为 x₂ − x₁，垂直差为 y₂ − y₁，因此距离等于两者平方和的平方根。
d = √((x₂ − x₁)² + (y₂ − y₁)²)
1, 2) and B(5, 5), the horizontal difference is 4 and the vertical difference is 3, so d = √16 + 9 = √25 = 5. This is a 3-4-5 triangle in disguise, which is a common pattern in exam questions.
对 A(1, 2) 与 B(5, 5)，水平差为 4，垂直差为 3，因此 d = √16 + 9 = √25 = 5。这实际上是一个 3-4-5 直角三角形，是考试题中常见的模式。
For A(2, 3) and B(6, 11), d = √16 + 64 = √80 = 4√5 form unless the question asks for a decimal, and round decimals to the accuracy stated in the question.
对 A(2, 3) 与 B(6, 11)，d = √16 + 64 = √80 = 4√5 ≈ 8.94。除题目要求小数外，应保留精确根式形式；若要求小数，则按题目规定的精度四舍五入。
The distance formula is also a useful tool for proving geometric facts. If two sides of a triangle have equal length, the triangle is isosceles; if the squares of the two shorter sides add to the square of the longest side, the triangle is right-angled.
距离公式也是证明几何结论的有用工具。若三角形两条边长度相等，则为等腰三角形；若较短两边的平方和等于最长边的平方，则为直角三角形。
3. Equation of a Straight Line | 直线方程
and c is the y-intercept, the value of y where the line crosses the y-axis. This form is ideal when the gradient and the y-intercept are easy to read from a graph or from given information.
直线最常见的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距，即直线与 y 轴交点的 y 值。当斜率与 y 轴截距能直接从图像或已知信息读出时，这种形式最方便。
y = m x + c
e not graphs of functions of x, since one x value corresponds to infinitely many y values.
水平直线的形式为 y = k，其中 k 为常数，因为其斜率为零。垂直直线的形式为 x = k，它不是 x 的函数图像，因为一个 x 值对应无穷多个 y 值。
constants. This form is often required in exam answers, especially when the question asks for integer coefficients with no common factor.
直线也可写成一般式 ax + by + c = 0，其中 a、b、c 为常数。考试答案常要求这种形式，尤其当题目要求整数系数且无公因数时。
get x = 4. Rearranging gives y = −¾ x + 3, so the gradient is −¾ and the y-intercept is 3.
求 3x + 4y = 12 的截距：令 x = 0，得 y = 3；令 y = 0，得 x = 4。整理得 y = −¾ x + 3，所以斜率为 −¾，y 轴截距为 3。
| Form | 形式 | Best used when | 适用情形 |
| y = mx + c | Gradient and y-intercept are known | 已知斜率与 y 轴截距 |
| y − y₁ = m(x − x₁) | Gradient and one point are known | 已知斜率与一个点 |
| ax + by + c = 0 | Required final form with integer coefficients | 要求整数系数的最终形式 |
4. Point-Gradient Form and Rearranging | 点斜式与变形
When you know the gradient m and one point (x₁, y₁) on the line, the point-gradient form gives the equation immediately. It is the single most useful formula in this chapter because almost every question can be reduced to finding a gradient and a point.
当已知斜率 m 与直线上一点 (x₁, y₁) 时，点斜式可立即给出方程。它是本章最有用的公式，因为几乎所有题目都可归结为求斜率和找一点。
y − y₁ = m(x − x₁)
x − 13. Always check by substituting the original point: 3(4) − 13 = −1, which is correct.
若直线过 (4, −1) 且斜率为 3，代入得 y − (−1) = 3(x − 4)，即 y + 1 = 3x − 12，所以 y = 3x − 13。务必用原点验证：3(4) − 13 = −1，正确。
er point and apply the point-gradient form. Choosing the point with simpler coordinates often reduces arithmetic errors.
若已知两点求方程，先用 m = (y₂ − y₁) / (x₂ − x₁) 求斜率，再任选一点使用点斜式。选择坐标较简单的一点通常能减少计算错误。
y − 5 = −2(x + 3) becomes y = −2x − 1, and then 2x + y + 1 = 0 with integer coefficients.
变形时先展开括号、合并同类项；若要求一般式，则把所有项移到一侧。例如 y − 5 = −2(x + 3) 化为 y = −2x − 1，再写成 2x + y + 1 = 0，系数均为整数。
c are integers. Multiply through by any denominator to clear fractions, and make sure the final expression has no common factor other than 1.
常见要求是把答案写成 ax + by + c = 0 的形式，其中 a、b、c 为整数。若有分母，先乘以公分母消去分数，并确保最终表达式除 1 外没有公因数。
5. Parallel and Perpendicular Lines | 平行与垂直直线
l lines never meet, and the vertical distance between them is constant. This makes parallel conditions useful for finding unknown constants in equations.
当两条直线斜率相同，即 m₁ = m₂ 时，它们平行。平行直线永不相交，且它们之间的垂直距离保持不变。因此平行条件常用于求方程中的未知常数。
= −1 / m₁. The perpendicular gradient is the negative reciprocal of the original gradient.
当两条直线斜率的乘积为 −1，即 m₁ × m₂ = −1 时，它们垂直，等价于 m₂ = −1 / m₁。垂直斜率与原斜率互为负倒数。
m₁ × m₂ = −1 ⇔ m₂ = −1 / m₁
rizontal line with gradient 0 is perpendicular to a vertical line with undefined gradient.
例如，斜率为 ⅔ 的直线与斜率为 −3/2 的直线垂直，因为 (2 / 3) × (−3 / 2) = −1。斜率为 0 的水平直线与斜率无定义的垂直直线互相垂直。
-gradient form. For the line y = 3x − 1 and the point (2, 5), the perpendicular gradient is −⅓, giving y − 5 = −⅓(x − 2) and finally y = −x / 3 + 17 / 3.
求过某点的垂直直线时，先求原直线斜率，取其负倒数，再用点斜式。对直线 y = 3x − 1 与点 (2, 5)，垂直斜率为 −⅓，得 y − 5 = −⅓(x − 2)，最终 y = −x / 3 + 17 / 3。
y = mx + c first, or use m = −a / b, before applying parallel or perpendicular conditions.
注意以 ax + by + c = 0 形式给出的直线。应先整理为 y = mx + c，或使用 m = −a / b，再应用平行或垂直条件。
6. Intersections of Lines | 直线的交点
The point of intersection of two straight lines is found by solving their equations simultaneously. The coordinates of the intersection satisfy both equations at the same time, so it is the single point shared by the two lines.
两条直线的交点通过联立它们的方程求解。交点坐标同时满足两个方程，因此它是两条直线唯一共有的点。
y = 5, so the intersection is (2, 5). Always substitute back into both equations to check.
对 y = 2x + 1 与 y = −x + 7，令右边相等：2x + 1 = −x + 7，得 3x = 6，x = 2。代入得 y = 5，所以交点为 (2, 5)。务必代回两个方程检验。
after isolating one variable or use elimination by adding or subtracting multiples of the equations. Both methods give the same answer; choose the one that keeps the numbers small.
当方程以 ax + by = c 形式给出时，可先解出一个变量再代入，或通过加减方程的倍数消元。两种方法结果相同，选择使数字更简单的一种。
are the same line. If solving leads to a contradiction such as 0 = 5, the lines are parallel; if it leads to an identity such as 0 = 0, the lines coincide.
两条直线可能恰有一个交点；若平行且不同，则无交点；若为同一直线，则有无穷多个交点。若求解得到 0 = 5 这样的矛盾，则两线平行；若得到 0 = 0 这样的恒等式，则两线重合。
ormed by lines and axes, you can calculate its area using ½ × base × perpendicular height.
交点问题常与面积问题结合。求出直线与坐标轴围成的三角形顶点后，可用 ½ × 底 × 高 计算面积。
7. Modelling with Straight Lines | 用直线建模
ere m is the rate of change and c is the initial value. In modelling questions, m and c must always be interpreted in the context of the problem, including their units.
线性模型的形式为 y = mx + c，其中 m 是变化率，c 是初始值。在建模题中，必须结合题目情境解释 m 与 c 的含义，并说明其单位。
80d, where C is the cost in pounds and d is the distance in kilometres, the value 3.50 is the fixed charge and 1.80 is the cost per kilometre. The gradient tells you how quickly the cost increases.
若出租车费用模型为 C = 3.50 + 1.80d，其中 C 为费用（英镑），d 为距离（千米），则 3.50 是固定收费，1.80 是每千米费用。斜率说明费用增长的速度。
here F is degrees Fahrenheit and C is degrees Celsius. Here the gradient 9 / 5 means each 1 °C increase corresponds to a 1.8 °F increase.
温度换算是另一个经典线性模型：F = (9 / 5)C + 32，其中 F 为华氏度，C 为摄氏度。这里斜率 9 / 5 表示每升高 1 °C 对应升高 1.8 °F。
Linear models are usually valid only over a sensible domain. A taxi fare model may not apply for negative distances, and a depreciation model may break down after many years. State any restrictions you notice, and avoid extrapolating far beyond the given data.
线性模型通常只在合理定义域内有效。出租车费用模型不适用于负距离，折旧模型在多年后也可能失效。指出你注意到的限制条件，并避免远超出已知数据进行外推。
When a question asks you to interpret the gradient, use a full sentence with units, such as the cost increases by 1.80 pounds for each additional kilometre travelled. This is worth a mark and is easy to lose.
当题目要求解释斜率时，用带单位的完整句子作答，例如每多行驶 1 千米，费用增加 1.80 英镑。这一分很容易拿到，也很容易丢失。
8. Sketching and Interpreting Graphs | 画图与解读图像
straight edge through them. A third point, such as the point where x = 1, acts as a check.
画直线草图时，找两个可靠的点，通常是 x 轴截距与 y 轴截距，然后用直尺连接。第三个点，例如 x = 1 时的点，可用作检验。
Sketching is not precise plotting. Label the intercepts and the equation of the line, use a sensible scale, and make the sign of the gradient visually correct: a positive gradient must clearly rise to the right.
草图不是精确描点。标出截距与直线方程，选择合理比例，并让斜率的符号在视觉上正确：正斜率必须明显向右上升。
To read a gradient from a graph, choose two points that lie on grid intersections, form a right-angled triangle, and divide the vertical change by the horizontal change. Use the scales on both axes carefully, because they may differ.
从图像读取斜率时，选两个位于网格交点上的点，构作直角三角形，用垂直变化量除以水平变化量。注意两个坐标轴的比例可能不同。
Interpreting a graph means translating its features into context. The y-intercept is often the starting value, the gradient is the rate of change, and the x-intercept is the value at which the quantity becomes zero.
解读图像意味着把图像特征翻译为情境含义。y 轴截距常表示初始值，斜率表示变化率，x 轴截距表示该量变为零时的取值。
You should also be able to recognise when a linear model is not appropriate. If a scatter graph curves clearly, fitting a straight line may still be possible as an approximation, but the model will only be reliable over a limited range.
你还应能识别线性模型何时不适用。若散点图明显弯曲，虽仍可拟合一条近似直线，但
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