📚 Mixed Exercise 5: Straight Line Graphs | 混合练习5：直线图像
This mixed exercise pulls together the key skills from Chapter 5 on straight line graphs. You will need to move fluently between gradients, equations of lines, parallel and perpendicular conditions, midpoints, distances and simple linear models. The questions are designed to test understanding rather than memory, so always show clear algebraic steps.
混合练习5综合了第5章“直线图像”的核心技能。你需要熟练地在斜率、直线方程、平行与垂直条件、中点、距离以及简单线性模型之间切换。这些题目考查的是理解而非记忆，因此务必展示清晰的代数步骤。
1. What Mixed Exercise 5 Covers | 混合练习5涵盖内容
The main topics include finding the gradient between two points, writing equations in different forms, using the relationship between parallel and perpendicular gradients, calculating midpoints and distances, and interpreting straight line models in context.
主要内容包括：求两点间的斜率、写出不同形式的直线方程、运用平行与垂直斜率的关系、计算中点与距离，以及在实际情境中解释直线模型。
- Gradient formula and its meaning
- Point-gradient form and y = mx + c
- Parallel and perpendicular line conditions
- Midpoint and distance between two points
- Linear models with real-life interpretation
- 斜率公式及其意义
- 点斜式与 y = mx + c 形式
- 平行线与垂直线的条件
- 中点与两点间距离
- 结合实际意义的线性模型
2. Gradient Between Two Points | 两点间的斜率
The gradient of a line passing through two points A(x₁, y₁) and B(x₂, y₂) measures how steep the line is. It is calculated as the change in y divided by the change in x.
经过两点 A(x₁, y₁) 和 B(x₂, y₂) 的直线斜率用来衡量直线的倾斜程度。它等于 y 的变化量除以 x 的变化量。
m = (y₂ – y₁) / (x₂ – x₁)
A positive gradient means the line rises from left to right, while a negative gradient means it falls. A zero gradient gives a horizontal line, and a vertical line has an undefined gradient.
正斜率表示直线从左到右上升，负斜率表示直线下降。斜率为零时直线水平，竖直直线的斜率不存在。
3. Equations of a Straight Line | 直线方程的形式
You need to be confident with two main forms. The point-gradient form is useful when you know one point and the gradient: y – y₁ = m(x – x₁). The gradient-intercept form is y = mx + c, where m is the gradient and c is the y-intercept.
你需要熟练掌握两种主要形式。点斜式适用于已知一个点和斜率的情况：y – y₁ = m(x – x₁)。斜截式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
y – y₁ = m(x – x₁) and y = mx + c
Questions may also ask for the general form ax + by + c = 0. You should be able to rearrange between these forms without losing accuracy.
题目也可能要求写成一般式 ax + by + c = 0。你应该能够在这些形式之间准确转换，不出现符号或移项错误。
4. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient, so m₁ = m₂. Perpendicular lines have gradients that multiply to give -1, so m₁ × m₂ = -1. This is often tested by asking for a line perpendicular to a given line through a specific point.
平行线的斜率相同，即 m₁ = m₂。垂直线的斜率乘积为 -1，即 m₁ × m₂ = -1。考试常考过某一点且与给定直线垂直的直线方程。
m₁ = m₂ for parallel; m₁ × m₂ = -1 for perpendicular
To find a perpendicular gradient, take the negative reciprocal of the original gradient. For example, if m = 2, the perpendicular gradient is -1/2.
求垂直斜率时，要取原斜率的负倒数。例如，若 m = 2，则垂直斜率为 -1/2。
5. Midpoints and Distances | 中点与距离
The midpoint of A(x₁, y₁) and B(x₂, y₂) is found by averaging the x-coordinates and the y-coordinates separately.
点 A(x₁, y₁) 与 B(x₂, y₂) 的中点坐标通过分别对 x 坐标和 y 坐标取平均值得到。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The distance between two points comes from Pythagoras’ theorem. It is the square root of the sum of the squared horizontal and vertical differences.
两点间的距离由勾股定理得出。它等于水平差与垂直差平方和的平方根。
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
These two tools are often combined when finding the equation of a perpendicular bisector.
这两项工具经常结合使用，例如求垂直平分线的方程。
6. Modelling with Straight Lines | 直线模型建模
Straight line models appear in real-life contexts such as cost, temperature or distance-time problems. The gradient represents a rate of change, and the y-intercept represents an initial or fixed value.
直线模型常出现在成本、温度或距离-时间等实际问题中。斜率表示变化率，y 轴截距表示初始值或固定值。
For example, if a company’s total cost C is given by C = 5x + 20, where x is the number of units produced, then 5 is the variable cost per unit and 20 is the fixed cost.
例如，若某公司的总成本 C 由 C = 5x + 20 给出，其中 x 为生产数量，则 5 表示每单位产品的变动成本，20 表示固定成本。
When answering modelling questions, always interpret the gradient and intercept in the context of the question, not just as abstract numbers.
在解答建模题时，务必结合题意解释斜率和截距的实际含义，而不仅仅把它们当作抽象数字。
7. Common Exam Mistake: Perpendicular Gradients | 常见考试失分点：垂直斜率
A very common error is to use the reciprocal of the gradient without changing the sign, or to change the sign without taking the reciprocal. The correct perpendicular gradient is always the negative reciprocal.
一个非常常见的错误是只取斜率的倒数而不改变符号，或只改变符号而不取倒数。正确的垂直斜率始终是负倒数。
For example, if the original gradient is 3/4, the perpendicular gradient is -4/3, not 4/3 or -3/4.
例如，若原斜率为 3/4，则垂直斜率为 -4/3，而不是 4/3 或 -3/4。
Another common mistake is mixing up the x- and y-coordinates when substituting into y – y₁ = m(x – x₁). Write the coordinates clearly before substitution.
另一个常见错误是在代入 y – y₁ = m(x – x₁) 时把 x 坐标和 y 坐标写混。代入前应清晰地写下坐标。
8. Worked Example 1: Finding the Equation | 例题1：求直线方程
Find the equation of the line passing through A(2, 5) and B(6, 13). First find the gradient: m = (13 – 5) / (6 – 2) = 8 / 4 = 2. Then use point-gradient form with A(2, 5): y – 5 = 2(x – 2). Simplifying gives y = 2x + 1.
求经过 A(2, 5) 和 B(6, 13) 的直线方程。首先求斜率：m = (13 – 5) / (6 – 2) = 8 / 4 = 2。然后用点 A(2, 5) 代入点斜式：y – 5 = 2(x – 2)。化简得 y = 2x + 1。
m = 2 → y = 2x + 1
You can check your answer by substituting both points into the final equation. For B(6, 13), 2(6) + 1 = 13, so the equation is correct.
你可以把两个点代入最终方程进行检验。对于 B(6, 13)，2(6) + 1 = 13，方程成立，因此答案正确。
9. Worked Example 2: Perpendicular Bisector | 例题2：垂直平分线
Find the perpendicular bisector of the line segment joining A(1, 2) and B(5, 6). The midpoint is M = ((1+5)/2, (2+6)/2) = (3, 4). The gradient of AB is (6 – 2) / (5 – 1) = 1, so the perpendicular gradient is -1.
求连接 A(1, 2) 和 B(5, 6) 的线段的垂直平分线方程。中点 M = ((1+5)/2, (2+6)/2) = (3, 4)。AB 的斜率为 (6 – 2) / (5 – 1) = 1，因此垂直斜率为 -1。
M = (3, 4) and m_perp = -1
Using point-gradient form through M(3, 4): y – 4 = -1(x – 3). Simplifying gives y = -x + 7, or x + y – 7 = 0 in general form.
用点 M(3, 4) 代入点斜式：y – 4 = -1(x – 3)。化简得 y = -x + 7，或写成一般式 x + y – 7 = 0。
10. Strategy for Mixed Exercise 5 | 混合练习5解题策略
Start by identifying what the question gives you: two points, one point and a gradient, or a parallel/perpendicular relationship to another line. Then choose the most efficient formula.
首先判断题目的已知条件：两个点、一个点和斜率，或者与另一条直线的平行/垂直关系。然后选择最有效的公式。
Write down the known values before substituting. If the question asks for a perpendicular bisector, find the midpoint first, then the perpendicular gradient, then the equation.
代入前先写下已知值。如果题目要求垂直平分线，先求中点，再求垂直斜率，最后求方程。
| Given information | Best starting step |
|---|---|
| Two points | Find gradient using m = (y₂ – y₁)/(x₂ – x₁) |
| Point and gradient | Use y – y₁ = m(x – x₁) |
| Parallel line | Keep the same gradient |
| Perpendicular line | Use negative reciprocal gradient |
Use the table to match the given information with the correct starting step. This reduces the chance of using the wrong formula under exam pressure.
利用表格将已知信息与正确的起始步骤对应起来。这可以降低考试压力下用错公式的概率。
11. Practice Checklist | 练习自查清单
Before attempting a mixed exercise, check that you can do all of the following quickly and accurately.
在开始混合练习前，检查你是否能快速准确地完成以下所有内容。
- Calculate gradient from two points
- Write y = mx + c from a graph or given information
- Find the equation of a parallel line
- Find the equation of a perpendicular line
- Calculate midpoints and distances
- Interpret gradient and intercept in a real-life model
- 由两点计算斜率
- 根据图像或已知信息写出 y = mx + c
- 求平行线的方程
- 求垂直线的方程
- 计算中点与距离
- 解释实际模型中斜率和截距的含义
If any item feels slow or uncertain, review the relevant section before doing the full mixed exercise. Accuracy with basic steps is more valuable than speed.
如果任何一项让你感到慢或不确定，在做完整的混合练习前先复习相关小节。基础步骤的准确性比速度更重要。
12. Final Summary | 本节总结
Mixed Exercise 5 tests your ability to combine coordinate geometry skills in one problem. The key formulas for gradient, point-gradient form, midpoint and distance must be automatic.
混合练习5考查你在同一道题中综合运用坐标几何技能的能力。斜率、点斜式、中点和距离的关键公式必须达到自动化的熟练程度。
Always watch for parallel and perpendicular conditions, and practise interpreting linear models in context. Careful substitution and checking your final equation with both points will help you avoid small but costly errors.
务必注意平行与垂直条件，并练习结合情境解释线性模型。仔细代入并用两个点检验最终方程，可以帮助你避免小而代价高昂的错误。
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