📚 Linear Equations and Straight-Line Graphs | 一次方程与直线图像
Linear equations and straight-line graphs form a core part of the Cambridge KS3 mathematics curriculum. They connect algebra with geometry and give students essential tools for solving problems in science, economics and everyday life.
一次方程与直线图像是剑桥 KS3 数学课程的核心内容。它们将代数与几何联系起来，为学生提供解决科学、经济和日常生活中问题的重要工具。
1. What is a Linear Equation? | 什么是一次方程？
e first power. There are no x² terms, no square roots of x, and no products such as x × x.
一次方程是指未知数（通常写作 x）只以一次幂出现的方程。方程中没有 x² 项，没有 x 的平方根，也没有 x × x 这样的乘积。
= c, where a, b and c are known numbers and a is not zero. Some examples are shown below.
一般形式为 ax + b = c，其中 a、b 和 c 是已知数，且 a 不等于零。下面是一些例子。
- 3x + 5 = 14 (linear, because the highest power of x is 1)
- 2(x - 4) = 10 (linear after expanding the bracket)
- x/3 - 2 = 7 (linear, because x is divided by 3, not squared)
例如 3x + 5 = 14 是线性方程，因为 x 的最高次数是 1；2(x - 4) = 10 展开后是线性方程；x/3 - 2 = 7 也是线性方程，因为 x 除以 3 而不是平方。
2. Solving One-Step Equations | 解一步一次方程
To solve a one-step linear equation, apply the inverse operation to both sides. This keeps the equation balanced, because whatever you do to one side, you must do to the other side.
解一步一次方程时，在等式两边同时进行逆运算。这样可以保持等式平衡，因为对一边做什么，对另一边也必须做什么。
For example, solve x + 5 = 12. Subtract 5 from both sides: x = 7.
例如，解 x + 5 = 12。两边同时减去 5：x = 7。
For multiplication, solve 4x = 20 by dividing both sides by 4: x = 5.
对于乘法，解 4x = 20 时，两边同时除以 4：x = 5。
| Operation | Inverse operation |
|---|---|
| + a | - a |
| - a | + a |
| × a | ÷ a |
| ÷ a | × a |
加法和减法互为逆运算，乘法和除法也互为逆运算。选择正确的逆运算可以快速求出未知数的值。
3. Solving Two-Step Equations | 解两步一次方程
equires two inverse operations. Usually, undo addition or subtraction first, then undo multiplication or division.
像 2x + 3 = 11 这样的两步方程需要两步逆运算。通常先消去加法或减法，再消去乘法或除法。
Example: solve 5x - 7 = 18.
例如：解 5x - 7 = 18。
Step 1: Add 7 to both sides: 5x = 25.
步骤 1：两边同时加 7：5x = 25。
Step 2: Divide both sides by 5: x = 5.
步骤 2：两边同时除以 5：x = 5。
This two-stage process is the same for most linear equations at KS3 level. Practise writing each step clearly to avoid arithmetic mistakes.
这个两步过程适用于 KS3 阶段大多数一次方程。练习时要把每一步写清楚，避免计算错误。
4. Equations with Brackets | 含括号的一次方程
ation contains brackets, expand them first using the distributive law: a(b + c) = ab + ac.
当方程含有括号时，先用分配律展开：a(b + c) = ab + ac。
solve 3(x - 2) = 15. Expand to 3x - 6 = 15, add 6 to get 3x = 21, then divide by 3: x = 7.
例如：解 3(x - 2) = 15。展开得 3x - 6 = 15，两边加 6 得 3x = 21，再除以 3：x = 7。
it back into the original equation. Here, 3(7 - 2) = 3 × 5 = 15, so the answer is correct.
务必把解代回原方程进行检验。这里 3(7 - 2) = 3 × 5 = 15，因此答案正确。
5. Equations with Unknowns on Both Sides | 两边都含未知数的方程
side. You can subtract 3x from both sides: 2x + 2 = 10, then subtract 2: 2x = 8, so x = 4.
对于像 5x + 2 = 3x + 10 这样的方程，将含 x 的项移到一边，数字项移到另一边。可以两边同时减去 3x：2x + 2 = 10，再减去 2：2x = 8，所以 x = 4。
step; the correct result with -2x = 8 is x = -4 only if the sign is handled consistently.
最好保持 x 的系数为正，但任何正确的平衡步骤都可以。如果你改为两边同时减去 5x，会得到 -2x + 2 = 10，然后 -2x = 8，最后 x = -4，这与前面的答案不同，说明符号处理出现错误，需要小心。
To avoid sign errors, check your working by substituting the final answer into the original equation.
为避免符号错误，请把最终答案代回原方程进行检验。
6. Solving Linear Inequalities | 解一次不等式
me way as equations, except that multiplying or dividing by a negative number reverses the inequality sign.
一次不等式使用 <、>、≤ 和 ≥ 等符号。它们的解法与方程相同，不同之处是乘以或除以负数时，不等号方向要改变。
Example: solve -2x < 8. Divide both sides by -2 and reverse the sign: x > -4.
例如：解 -2x < 8。两边同时除以 -2 并改变不等号方向：x > -4。
e solution on a number line, use an open circle for < or > and a closed circle for ≤ or ≥.
在数轴上表示解时，< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆圈。
7. Coordinate Plane and Plotting Points | 坐标平面与描点
The coordinate plane is made of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). A point is written as an ordered pair (x, y).
坐标平面由两条相互垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。一个点写成有序数对 (x, y)。
The first number tells you how far to move left or right from the origin (0, 0), and the second number tells you how far to move up or down.
第一个数表示从原点 (0, 0) 向左或向右移动多远，第二个数表示向上或向下移动多远。
For example, the point (3, -2) means move 3 units to the right and 2 units down. The point (-4, 5
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