📚 Master Straight-Line Graphs: y = mx + c for Cambridge KS3 Maths | 掌握直线图像：剑桥KS3数学中的 y = mx + c
ribes a line on a coordinate grid. This article builds your understanding step by step, from plotting points to finding gradients and intercepts.
直线图像是KS3数学中最重要的概念之一。它将代数与几何联系起来，说明像 y = mx + c 这样的方程如何在坐标网格上描述一条直线。本文将循序渐进地帮助你建立从描点到求斜率和截距的理解。
1. What Is a Linear Equation? | 什么是线性方程？
equation that can be written in the form y = mx + c is called a linear equation. The word 'linear' ns means that when you draw all the points that satisfy the equation, they form a straight line.
任何可以写成 y = mx + c 形式的方程都称为线性方程。“线性”一词表示当你画出所有满足该方程的点时，它们会形成一条直线。
The highest power of x in a linear equation is 1. This means there are no x² terms, no x³ terms, and no fractions where x is in the denominator.
线性方程中 x 的最高次数是 1。这意味着没有 x² 项、没有 x³ 项，也没有 x 在分母中的分数。
5 and y = ½x − 1 are all linear equations. However, y = x² + 1 and y = 1/x are not linear.
例如，y = 2x + 3、y = −x + 5 和 y = ½x − 1 都是线性方程。然而，y = x² + 1 和 y = 1/x 不是线性的。
y = mx + c
Here, m is the gradient and c is the y-intercept. Understanding these two numbers is the key to drawing and describing straight lines.
这里，m 是斜率，c 是 y 轴截距。理解这两个数字是绘制和描述直线的关键。
2. Plotting Points and Drawing Lines | 描点与画直线
To draw a straight line, you only need two points, but using a table of values with three or more points makes your graph more reliable.
要画一条直线，你只需要两个点，但使用包含三个或更多点的数值表可以使图像更可靠。
Choose a set of x-values, substitute each one into the equation, and find the matching y-value. Then plot each pair as a coordinate (x, y).
选择一组 x 值，将每个值代入方程，求出对应的 y 值。然后将每一对数值作为坐标 (x, y) 描点。
2x + 1, we can choose x = −2, −1, 0, 1, 2. The table below shows the calculated y-values.
对于 y = 2x + 1，我们可以选择 x = −2、−1、0、1、2。下表显示了计算出的 y 值。
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = 2x + 1 | −3 | −1 | 1 | 3 | 5 |
Plot the points (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). Then draw a single straight line through all of them.
描出点 (−2, −3)、(−1, −1)、(0, 1)、(1, 3) 和 (2, 5)。然后画一条穿过所有点的直线。
3. The Gradient m: Rise over Run | 斜率m：纵增量除以横增量
The gradient of a line measures how steep it is. It tells you how much y changes when x increases by 1.
直线的斜率衡量它的倾斜程度。它告诉你当 x 增加 1 时，y 变化了多少。
m = Δy / Δx = rise / run
Here, Δy is the vertical change and Δx is the horizontal change between two points.
这里，Δy 是两点之间的纵变化，Δx 是横变化。
If m is positive, the line slopes upwards from left to right. If m is negative, the line slopes downwards from left to right.
如果 m 为正，直线从左到右向上倾斜。如果 m 为负，直线从左到右向下倾斜。
the gradient is 3. This means for every 1 unit you move right, the line moves 3 units up.
对于直线 y = 3x − 2，斜率为 3。这意味着每向右移动 1 个单位，直线向上移动 3 个单位。
4. The y-intercept c: Where the Line Meets the y-axis | y截距c：直线与y轴的交点
intercept is the value of y when x = 0. It is the point where the line crosses the y-axis.
y 轴截距是当 x = 0 时 y 的值。它是直线与 y 轴相交的点。
In the equation y = mx + c, the constant c is always the y-intercept.
在方程 y = mx + c 中，常数 c 始终是 y 轴截距。
line crosses the y-axis at (0, 5). In y = −x − 3, the line crosses the y-axis at (0, −3).
例如，在 y = 2x + 5 中，直线在 (0, 5) 处与 y 轴相交。在 y = −x − 3 中，直线在 (0, −3) 处与 y 轴相交。
t can be positive, negative, or zero. If c = 0, the line passes through the origin (0, 0).
y 轴截距可以是正数、负数或零。如果 c = 0，直线经过原点 (0, 0)。
5. Reading y = mx + c | 解读 y = mx + c
Once you know the parts of the equation, you can quickly describe any straight line without drawing a table of values.
一旦你了解了方程的各个部分，就可以快速描述任何直线，而无需绘制数值表。
Look at the number in front of x: that is the gradient m. Look at the constant term on its own: that is the y-intercept c.
看 x 前面的数字：那是斜率 m。看单独的常数项：那是 y 轴截距 c。
| Equation | Gradient m | y-intercept c |
|---|---|---|
| y = 4x − 7 | 4 | −7 |
| y = −2x + 6 | −2 | 6 |
| y = ¾x | ¾ | 0 |
− 3x, the gradient is −3 and the y-intercept is 5, because it is the same as y = −3x + 5.
注意：如果方程写作 y = 5 − 3x，那么斜率为 −3，y 轴截距为 5，因为它与 y = −3x + 5 相同。
6. Finding the Gradient from Two Points | 从两点求斜率
You do not always need a full equation to find the gradient. If you know two points on a line, use the formula below.
你并不总是需要完整的方程来求斜率。如果你知道直线上的两个点，可以使用下面的公式。
m = (y₂ − y₁) / (x₂ − x₁)
Here, (x₁, y₁) and (x₂, y₂) are any two different points on the line.
这里，(x₁, y₁) 和 (x₂, y₂) 是直线上任意两个不同的点。
Example: Find the gradient of the line passing through (2, 5) and (4, 11).
示例：求经过 (2, 5) 和 (4, 11) 的直线的斜率。
Substitute the values: m = (11 − 5) / (4 − 2) = 6 / 2 = 3.
代入数值：m = (11 − 5) / (4 − 2) = 6 / 2 = 3。
Always subtract the y-values and x-values in the same order to avoid a sign error.
始终以相同顺序相减 y 值和 x 值，以避免符号错误。
7. Finding the Equation of a Straight Line | 求直线方程
e gradient and one point on a line, you can find the full equation in the form y = mx + c.
如果你知道直线的斜率和直线上的一个点，就可以求出 y = mx + c 形式的完整方程。
ing the known gradient and the coordinates of the point into y = mx + c. Then solve for c.
首先将已知斜率和点的坐标代入 y = mx + c。然后解出 c。
Example: A line has gradient 2 and passes through (3, 8). Find its equation.
示例：一条直线的斜率为 2，并且经过点 (3, 8)。求它的方程。
Substitute m = 2, x = 3, y = 8 into y = mx + c: 8 = 2 × 3 + c, so 8 = 6 + c, giving c = 2.
将 m = 2、x = 3、y = 8 代入 y = mx + c：8 = 2 × 3 + c，所以 8 = 6 + c，得到 c = 2。
The equation is therefore y = 2x + 2.
因此方程是 y = 2x + 2。
8. Parallel Lines and Their Gradients | 平行线及其斜率
Parallel lines never meet. They always have the same gradient but different y-intercepts.
平行线永不相交。它们始终具有相同的斜率，但 y 轴截距不同。
must also have gradient 3. For example, y = 3x − 4 and y = 3x + 7 are both parallel to it.
如果一条直线的方程是 y = 3x + 1，那么任何与它平行的直线也必须有斜率 3。例如，y = 3x − 4 和 y = 3x + 7 都与它平行。
If two lines have different gradients, they will intersect at exactly one point. If two lines have the same gradient and the same y-intercept, they are the same line.
如果两条直线的斜率不同，它们将恰好相交于一点。如果两条直线的斜率相同且 y 轴截距也相同，它们就是同一条直线。
Checking gradients is a useful way to decide whether lines are parallel in exam questions.
在考试题目中，检查斜率是判断直线是否平行的一种有用方法。
9. Using Graphs to Solve Problems | 利用图像解决问题
Straight-line graphs can model real situations, such as mobile phone costs, temperature changes, or distance travelled over time.
直线图像可以模拟实际情况，例如手机费用、温度变化或距离随时间的变化。
When two straight lines are drawn on the same set of axes, their point of intersection is the solution to both equations.
当两条直线画在同一坐标系上时，它们的交点就是两个方程的共同解。
+ 2 and y = −x + 6 intersect at (2, 4). This means x = 2 and y = 4 satisfy both equations.
示例：y = x + 2 和 y = −x + 6 的图像相交于点 (2, 4)。这意味着 x = 2 和 y = 4 同时满足两个方程。
You can also read values from a straight-line graph by moving vertically from the x-axis to the line, then horizontally to the y-axis.
你也可以通过从 x 轴垂直移动到直线，再水平移动到 y 轴来读出直线图上的数值。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many marks are lost through small errors. Here are some common mistakes to avoid when working with linear graphs.
许多分数是因为小错误而丢失的。以下是在处理线性图像时需要避免的一些常见错误。
cept. In y = 5x − 3, the gradient is 5 and the y-intercept is −3, not the other way round.
不要将斜率与 y 轴截距混淆。在 y = 5x − 3 中，斜率为 5，y 轴截距为 −3，而不是相反。
When calculating the gradient from two points, make sure you subtract the coordinates in the same order for both y and x.
从两点计算斜率时，请确保 y 和 x 的坐标按相同顺序相减。
Always label your axes and use a ruler to draw a straight line through the plotted points. Check that the line passes through all your calculated coordinates.
始终标注坐标轴，并使用直尺在描出的点上画直线。检查直线是否经过你计算的所有坐标。
c, rearrange it first. For example, 2y = 6x + 4 becomes y = 3x + 2, so the gradient is 3.
最后，如果方程不是 y = mx + c 形式，请先重新整理。例如，2y = 6x + 4 可化为 y = 3x + 2，因此斜率为 3。
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