📚 Mastering IGCSE Linear Equations and Straight-Line Graphs | 掌握 IGCSE 线性方程与直线图像
Linear relationships are everywhere in IGCSE Mathematics, from mobile phone bills to speed-time problems. This article brings together solving linear equations, inequalities, rearranging formulae and straight-line graphs. Master these skills and you will be ready for Core and Extended questions that regularly carry high marks.
线性关系在 IGCSE 数学中无处不在，从手机话费到速度-时间问题。本文汇总解一元一次方程、不等式、公式变形以及直线图像。掌握这些技能，你就能应对核心和扩展试卷中经常出现的高分值题目。
1. Linear Equations: The Foundation | 线性方程：基础
oth sides: subtract b, then divide by a. For example, 3x + 5 = 20 gives 3x = 15, so x = 5.
一元一次方程的一般形式是 ax + b = c，其中 x 是未知数。目标是利用逆运算把 x 单独留在一边。解题时两边必须同时进行相同操作：先减去 b，再除以 a。例如 3x + 5 = 20，得到 3x = 15，所以 x = 5。
therefore x = 4. Always check by substituting the answer back into the original equation.
如果两边都有 x，要先合并同类项。解 5x − 2 = 2x + 10，两边同时减去 2x 并加上 2：得到 3x = 12，因此 x = 4。一定要把答案代回原方程进行检验。
3x + 5 = 20 → 3x = 15 → x = 5
2. Word Problems into Equations | 从文字题到方程
Many IGCSE questions ask you to form an equation from a written situation. Define the unknown clearly, often with the phrase 'Let x be ...'. Then translate words into algebra: 'three more than double a number' becomes 2x + 3.
许多 IGCSE 题目要求根据文字情境建立方程。先明确定义未知数，通常使用 'Let x be ...' 这样的表达。然后把文字翻译成代数式：例如 '一个数的两倍再加三' 写成 2x + 3。
2x + 2(x + 5) = 46. Expanding gives 4x + 10 = 46, so x = 9. Therefore the length is 14 cm.
例题：一个矩形的周长是 46 cm，长比宽多 5 cm。设宽为 x cm，那么长为 x + 5 cm。周长方程为 2x + 2(x + 5) = 46。展开得 4x + 10 = 46，所以 x = 9。因此长为 14 cm。
2x + 2(x + 5) = 46 → x = 9
3. Inequalities on a Number Line | 数轴上的不等式
uality x ≤ −2 means all values less than or equal to −2, shown with a closed circle at −2.
不等式用 <、>、≤ 或 ≥ 来比较大小。x > 3 表示所有大于 3 的数，在 3 处画空心圆。x ≤ −2 表示所有小于或等于 −2 的数，在 −2 处画实心圆。
x ≤ 4, shade the region between 1 and 4, use an open circle at 1 and a closed circle at 4.
在数轴上箭头方向必须正确。对于复合不等式如 1 < x ≤ 4，在 1 和 4 之间涂色，1 处画空心圆，4 处画实心圆。
x > 3: open circle at 3, arrow to the right
4. Solving Linear Inequalities | 解一元一次不等式
Inequalities are solved in the same way as equations, with one vital exception: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. Solve 5 − 2x > 1: subtract 5 to get −2x > −4; then divide by −2 and reverse the sign to obtain x < 2.
解不等式与解方程的方法相同，但有一个关键例外：如果两边同时乘以或除以一个负数，必须反转不等号。解 5 − 2x > 1：先减 5 得 −2x > −4；然后除以 −2 并反转符号，得到 x < 2。
The answer can be shown on a number line or written as an inequality. IGCSE mark schemes usually accept x < 2 or the equivalent graph. Always decide whether the circle should be open or closed by checking the original symbol.
答案可以表示在数轴上，也可以写成不等式形式。IGCSE 评分标准通常接受 x < 2 或等价的数轴图示。圈是空心还是实心，要根据原不等式中的符号来判断。
5 − 2x > 1 → −2x > −4 → x < 2
5. Rearranging Formulae | 公式变形
Treat the new subject as the unknown quantity and move everything else to the other side.
改变公式的主项时，使用与解方程相同的逆运算方法。若 v = u + at，把 t 作为主项，得到 t = (v − u) ÷ a。把新主项当作未知量，把其他量都移到等号的另一边。
3)/2, then 2y = x + 3, so x = 2y − 3. This removes the fraction before you collect terms.
公式中含有分数时，先两边同乘分母。若 y = (x + 3)/2，则 2y = x + 3，所以 x = 2y − 3。这样可以先消去分数，再合并同类项。
v = u + at → t = (v − u) ÷ a
6. y = mx + c: The Straight-Line Form | y = mx + c：直线方程形式
aight line, where m is the gradient and c is the y-intercept. The gradient measures steepness; the intercept is the point where the line crosses the y-axis. This form is essential for sketching graphs and for reading off the gradient directly.
方程 y = mx + c 表示一条直线，其中 m 是斜率，c 是 y 轴截距。斜率衡量直线的倾斜程度；截距是直线与 y 轴的交点。这个形式对于画直线图像和直接读出斜率都至关重要。
3); then rise 2 units and run 1 unit to plot a second point. Join the points with a ruler.
画直线图像时，先标出 y 轴截距，再把斜率看作纵向增量除以横向增量。对于 y = 2x − 3，从 (0, −3) 出发，纵向上升 2 个单位，横向移动 1 个单位，得到第二个点。然后用直尺连接两点。
y = mx + c
7. Gradient from Two Points | 两点求斜率
2) ÷ (4 − 1) = 9 ÷ 3 = 3. The gradient is positive because the line rises as x increases.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率用公式 m = (y₂ − y₁) ÷ (x₂ − x₁) 计算。对于 (1, 2) 和 (4, 11)，m = (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3。因为 x 增大时直线上升，所以斜率为正。
nt m = 0; a vertical line has an undefined gradient because the horizontal change is zero.
如果直线向下倾斜，斜率为负。水平线的斜率 m = 0；竖直线由于横向变化为零，斜率无定义。
m = (y₂ − y₁) ÷ (x₂ − x₁)
8. Parallel and Perpendicular Lines | 平行线与垂线
re parallel because both have m = 3. Their y-intercepts are different, so they never meet.
平行线斜率相等。例如 y = 3x + 1 与 y = 3x − 5 平行，因为两者的斜率都是 m = 3。它们的 y 轴截距不同，所以永远不会相交。
. This can be written as m₁ × m₂ = −1, where m₁ and m₂ are the gradients of the two lines.
两条垂线的斜率乘积为 −1。如果一条线的斜率 m = 2，那么与它垂直的直线斜率为 −1/2。这可以写作 m₁ × m₂ = −1，其中 m₁ 和 m₂ 是两条直线的斜率。
m₁ × m₂ = −1
9. Solving Simultaneous Equations Graphically | 图像法解
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