📚 Student's Edition Grade 1 Lesson 16-6: Equation of a Straight Line | 学生版第1级第16-6课：直线方程
The equation of a straight line is one of the most useful tools in coordinate geometry. It connects algebra to shapes on a graph and lets you solve problems about gradients, intercepts, parallel lines and perpendicular lines. This lesson guides you through every key method step by step, from reading coordinates to applying equations in real contexts.
直线方程是坐标几何中最有用的工具之一。它把代数与图像上的图形联系起来，让你能够解决涉及斜率、截距、平行线和垂直线的问题。本课将逐步带你掌握每一个关键方法，从读取坐标到在真实情境中应用方程。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
Every point on a flat surface can be described by an ordered pair (x, y). The x-coordinate tells you how far to move left or right from the origin, and the y-coordinate tells you how far to move up or down.
平面上的每一个点都可以用有序数对 (x, y) 来描述。x坐标表示从原点向左或向右移动的距离，y坐标表示向上或向下移动的距离。
The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. The origin is the point (0, 0) where the two axes meet. The plane is divided into four quadrants, but for straight line work you will mostly focus on the way points are plotted.
水平轴称为x轴，垂直轴称为y轴。原点是两个轴相交的点 (0, 0)。平面被分为四个象限，但在直线问题中，你主要关注点的描画方式。
When you plot points such as (2, 3) and (-1, 4), you can begin to see the shape of a line or curve. A straight line is produced by points that have a constant rate of change between x and y.
当你描出 (2, 3) 和 (-1, 4) 等点时，就能开始看出直线或曲线的形状。当点与点之间 x 和 y 的变化率恒定时，就会形成一条直线。
- Origin: the fixed starting point (0, 0) / 原点：固定起点 (0, 0)
- x-axis: horizontal, y = 0 along it / x轴：水平方向，线上 y = 0
- y-axis: vertical, x = 0 along it / y轴：垂直方向，线上 x = 0
- Quadrants: four regions numbered anticlockwise from the top right / 象限：从右上角开始逆时针编号的四个区域
2. What is a Straight Line Graph? | 什么是直线图像
A straight line graph is a set of points that all lie on one single straight line. Any two of those points are enough to draw the whole line, but the line itself contains infinitely many points. What makes the graph straight is that y changes by the same amount every time x increases by 1.
直线图像是指所有点都落在同一条直线上的图形。取其中任意两个点就足以画出整条直线，但这条直线本身包含无穷多个点。图像之所以是直的，是因为每当 x 增加 1，y 的变化量都相同。
e gradient and c is the y-intercept. This single equation describes the whole infinite line, which is why it is so powerful.
直线图像总可以写成 y = mx + c 的形式，其中 m 是斜率，c 是y截距。这一个方程就描述了整条无限的直线，这正是它如此强大的原因。
y = m x + c
se the rate of change is not constant. Recognising this difference is a common exam skill.
并非所有含 x 和 y 的方程都表示直线。例如 y = x² 和 y = 1 ÷ x 都是曲线，因为它们的变化率不恒定。能识别这一区别是考试中的常见技能。
3. Gradient: Measuring Steepness | 斜率：衡量倾斜程度
The gradient of a line measures how steep it is and in which direction it slopes. It is the vertical change divided by the horizontal change between any two points on the line. In words: gradient equals rise over run.
直线的斜率衡量它有多陡，以及朝哪个方向倾斜。它等于直线上任意两点之间的垂直变化量除以水平变化量。用文字说就是：斜率 = 竖直上升 ÷ 水平移动。
m = (y₂ − y₁) ÷ (x₂ − x₁)
the line rises from left to right, m is positive; if it falls, m is negative; if it is flat, m = 0; if it is vertical, m is undefined.
在直线上选取两个标记清楚的点和 (x₁, y₁) 与 (x₂, y₂)，按同样的顺序把 y 值相减、x 值相减，然后相除。若直线从左到右上升，m 为正；若下降，m 为负；若水平，m = 0；若竖直，m 不存在。
| Line direction / 直线方向 | Gradient / 斜率 | Example / 例子 |
| Rising left to right / 从左向右上升 | Positive / 正 | m = 3 |
| Falling left to right / 从左向右下降 | Negative / 负 | m = −2 |
| Horizontal / 水平 | Zero / 零 | m = 0 |
| Vertical / 竖直 | Undefined / 不存在 | x = 4 |
4. The y-intercept and y = mx + c | y截距与 y = mx + c
mx + c, the constant c is exactly that crossing value. This is why the form is so convenient: you can read the gradient and the intercept straight from the equation.
y截距是直线与y轴相交处 y 的值，而交点处的 x 一定等于 0。在方程 y = mx + c 中，常数 c 正是这个交点值。这正是该形式方便的原因：你可以直接从方程读出斜率与截距。
is at (0, −7). If you are ever unsure, substitute x = 0 into the equation and solve for y.
例如在 y = 4x − 7 中，斜率是 4，直线与y轴交于点 (0, −7)。如果不确定，只需把 x = 0 代入方程解出 y。
s gives 0 = 4x − 7, so x = 7 ÷ 4 = 1.75, meaning the line crosses the x-axis at (1.75, 0).
若要求x截距，则把 y = 0 代入并解出 x。对于 y = 4x − 7，得到 0 = 4x − 7，所以 x = 7 ÷ 4 = 1.75，即直线与x轴交于 (1.75, 0)。
5. Finding the Equation from a Graph | 从图像求方程
hoose two clear points and work out the gradient m. Third, write the answer as y = mx + c.
当直线已经画好时，你可以用三个快速步骤还原它的方程。第一步，标出y截距得到 c。第二步，选取两个清楚的点算出斜率 m。第三步，把答案写成 y = mx + c。
- Step 1: Read where the line cuts the y-axis / 第一步：读出直线与y轴的交点
- Step 2: Count the rise and the run between two lattice points / 第二步：数出两个格点之间的上升与水平移动
- Step 3: Substitute m and c into y = mx + c and check one point / 第三步：把 m 和 c 代入 y = mx + c 并用一个点检验
Always test your equation with a third point that you did not use. If the coordinates satisfy your equation, your answer is almost certainly correct, which also protects you against sign errors.
一定要用第三个你没用过的点来检验方程。如果该坐标满足你的方程，答案几乎肯定正确，这也能帮你避免符号错误。
6. Finding the Equation from Two Points | 由两点求方程
e the gradient formula first, then substitute one of the points into y = mx + c to find c.
如果题目只给两个点而没有图像，你同样可以求出方程。先用斜率公式求出 m，再把其中一个点代入 y = mx + c 求出 c。
Worked example / 例题: find the equation of the line through (2, 3) and (6, 11).
例题：求经过 (2, 3) 和 (6, 11) 的直线方程。
d c = −1. Step 3: the equation is y = 2x − 1. Check with (6, 11): 2 × 6 − 1 = 11, correct.
第一步：m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2。第二步：代入 (2, 3)：3 = 2 × 2 + c，得 3 = 4 + c，所以 c = −1。第三步：方程为 y = 2x − 1。用 (6, 11) 检验：2 × 6 − 1 = 11，正确。
Keep the order of subtraction consistent. If you subtract the y-values as y₂ − y₁, you must subtract the x-values as x₂ − x₁, otherwise the sign of your gradient will flip.
相减的顺序要保持一致。如果你把 y 值写作 y₂ − y₁，那么 x 值也必须写作 x₂ − x₁，否则斜率的符号会反过来。
7. Point-Gradient Form and Rearranging | 点斜式与方程变形
here (x₁, y₁) is the known point. Expanding and collecting terms turns it into y = mx + c.
当你知道斜率和一点时，点斜式能节省时间。它写作 y − y₁ = m(x − x₁)，其中 (x₁, y₁) 是已知点。展开并整理同类项后，就变成 y = mx + c。
ribe the same straight line, so either is acceptable unless the question specifies a form.
例如，斜率为 3 且经过 (4, 5) 的直线：y − 5 = 3(x − 4)，即 y − 5 = 3x − 12，所以 y = 3x − 7。两种形式描述的是同一条直线，除非题目指定形式，否则都可以。
+ 3y = 12, subtract 2x and divide by 3 to get y = −(2 ÷ 3)x + 4, so m = −2 ÷ 3 and c = 4.
有些考题给出的方程形式是 ax + by + c = 0。要读出斜率，就把它整理成 y 的表达式。对于 2x + 3y = 12，两边减去 2x 再除以 3，得 y = −(2 ÷ 3)x + 4，所以 m = −2 ÷ 3，c = 4。
8. Horizontal and Vertical Lines | 水平线与垂直线
the rise is zero and the gradient is 0. A horizontal line through (0, 5) is simply y = 5.
水平线的方程是 y = k，其中 k 是常数。直线上每个点的 y 值都相同，所以上升量为零，斜率为 0。经过 (0, 5) 的水平线就是 y = 5。
. Every point has the same x-value, so the run is zero. Dividing by zero is undefined, so the gradient of a vertical line is undefined rather than infinite in a usable sense.
垂直线的方程是 x = k。直线上每个点的 x 值都相同，所以水平移动量为零。除以零没有意义，因此竖直直线的斜率不存在，而不是在可用意义上的"无穷大"。
ight up and down. Mixing these two up is one of the most frequent careless errors in coordinate geometry.
一个好记的方法是：y = k 像地平线一样平，而 x = k 是笔直上下的。把这两者弄混是坐标几何中最常见的不小心错误之一。
9. Parallel Lines | 平行线
radients: if m₁ = m₂, the lines are parallel, and if the lines are parallel, then m₁ = m₂.
平行线永不相交，这意味着它们的上升速率完全相同。因此平行线的斜率相等：若 m₁ = m₂，则两直线平行；反之，若两直线平行，则 m₁ = m₂。
l to y = 2x + 1 passing through (3, 4) has m = 2, so 4 = 2 × 3 + c gives c = −2 and the equation is y = 2x − 2.
求平行线方程的方法：保持斜率不变，用新的点求出不同的截距。例如，与 y = 2x + 1 平行且经过 (3, 4) 的直线，m = 2，由 4 = 2 × 3 + c 得 c = −2，方程为 y = 2x − 2。
Notice that only the constant changed. If two equations have the same m but the same c as well, they are not two lines but one identical line, so always check whether the intercepts actually differ.
注意只有常数变了。如果两个方程的 m 相同、c 也相同，那就不是两条线而是同一条线，因此要始终检查截距是否真的不同。
10. Perpendicular Lines | 垂直相交的直线
Two lines are perpendicular when they meet at a right angle of 90°. For perpendicular lines the product of the gradients is −1, which means each gradient is the negative reciprocal of the other.
当两条直线相交成 90° 直角时，它们互相垂直。对于垂直的两条直线，斜率之积为 −1，也就是说每个斜率都是另一个斜率的负倒数。
m₁ × m₂ = −1 so m₂ = −1 ÷ m₁
ne line has gradient −3 ÷ 4, the perpendicular gradient is 4 ÷ 3. Remember to flip the fraction and change the sign.
若一条直线的斜率是 2，则垂直直线的斜率是 −1 ÷ 2 = −½。若一条直线的斜率是 −3 ÷ 4，则垂直斜率为 4 ÷ 3。记住要"取倒数并变号"。
Two special cases are worth memorising: a horizontal line is perpendicular to a vertical line, and the gradient of a horizontal line is 0 while the gradient of a vertical line is undefined. The product rule does not apply here because undefined is not a number.
有两个特例值得记住：水平线与竖直线互相垂直；水平线的斜率为 0，而竖直线的斜率不存在。这里乘积规则不适用，因为"不存在"不是一个数。
11. Real-Life Applications | 实际应用
re C is the total cost in pounds and n is the number of miles travelled. The gradient 2 is the cost per mile and the intercept 3 is the fixed charge.
线性方程可以描述任何以恒定速率变化的事物。例如出租车费用可写成 C = 3 + 2n，其中 C 是以英镑计的总价，n 是行驶的英里数。斜率 2 是每英里费用，截距 3 是起步价。
Other contexts include converting between units, predicting total savings over time, and modelling distance travelled at constant speed. In every case the same structure appears: a fixed starting amount plus a constant rate multiplied by the variable.
其他情境还包括单位换算、预测随时间累积的储蓄总额，以及以匀速行驶的距离建模。每种情形都呈现同样的结构：一个固定的起始量，加上恒定速率乘以变量。
- Fixed cost or starting value → the y-intercept c / 固定费用或起始值 → y截距 c
- Rate per unit, per hour or per item → the gradient m / 每单位、每小时或每件的费率 → 斜率 m
- Total or final value → the y of the equation / 总量或最终值 → 方程中的 y
When you answer a worded question, always state what your variables mean and give the final equation in the requested form, together with the units where appropriate.
回答文字应用题时，一定要说明各变量的含义，并按题目要求的形式给出最终方程，必要时带上单位。
12. Common Mistakes and Revision Checklist | 常见错误与复习清单
forgetting to change the sign for the negative reciprocal, and confusing x = k with y = k.
本主题失分大多来自少数几个反复出现的疏忽：减法顺序颠倒、求负倒数时忘记变号，以及把 x = k 与 y = k 混淆。
t is actually (5, 2). Slow down on the substitution step and write the coordinates clearly before you begin.
另一个常见错误是把点代错位置，例如点其实是 (5, 2)，却写成 5 = m × 2 + c。在代入步骤放慢速度，动手前把坐标写清楚。
- Can you plot and read coordinates in all four quadrants? / 你能在四个象限中描点并读坐标吗？
- Can you calculate a gradient from a graph and from two points? / 你能从图像和由两点求斜率吗？
- Can you find c by substituting a known point? / 你能通过代入已知点求出 c 吗？
- Can you state the gradients of parallel and perpendicular lines? / 你能说出平行线与垂直线的斜率关系吗？
- Can you recognise y = k and x = k on sight? / 你能一眼认出 y = k 与 x = k 吗？
- Can you apply a linear model to a real context with units? / 你能把线性模型应用到带单位的真实情境吗？
Work through past paper questions on straight lines until the three-step method becomes automatic. Accuracy in this lesson pays off later, because gradients and equations reappear in calculus, coordinate geometry and mechanics.
把直线相关的历年真题反复练到三步法成为习惯。本课的准确性会在之后带来回报，因为斜率与方程会在微积分、坐标几何和力学中再次出现。
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