📚 IGCSE Maths: Equations, Inequalities and Graphs | IGCSE 数学：方程、不等式与图像
This revision guide covers the core IGCSE Mathematics skills for solving linear and quadratic equations, simultaneous equations, inequalities, and interpreting graphs. Mastering these techniques is essential for both the Core and Extended 0580 syllabus.
本复习指南涵盖 IGCSE 数学中求解一次方程和二次方程、联立方程、不等式以及图像解读的核心技能。掌握这些方法对 0580 大纲的核心课程和拓展课程都至关重要。
1. Solving Linear Equations | 解一次方程
To solve a linear equation, collect like terms and isolate the unknown variable on one side. Always perform the same operation on both sides of the equation.
要解一次方程，先合并同类项，将未知数移到方程的一边。对方程两边必须同时进行相同的运算。
Example: Solve 3x + 5 = 2x + 12.
例如：解方程 3x + 5 = 2x + 12。
3x - 2x = 12 - 5 → x = 7
k by substituting x = 7 back into the original equation: 3(7) + 5 = 26 and 2(7) + 12 = 26.
将 x = 7 代回原方程检验：3(7) + 5 = 26，而 2(7) + 12 = 26，两边相等。
- 2, which simplifies to 2x - 2 = 5x - 2, then -2 + 2 = 5x - 2x, giving 0 = 3x, so x = 0.
当方程含有括号或分数时，先化简。例如，2(x - 3) + 4 = 5x - 2 展开为 2x - 6 + 4 = 5x - 2，化简得 2x - 2 = 5x - 2，再移项得 -2 + 2 = 5x - 2x，即 0 = 3x，所以 x = 0。
s all real numbers; if it is a contradiction, such as x + 1 = x - 1, there is no solution.
最终答案要书写清晰。如果方程是恒等式，如 2x + 4 = 2(x + 2)，解为全体实数；如果是矛盾式，如 x + 1 = x - 1，则无解。
2. Simultaneous Linear Equations | 联立一次方程
Simultaneous equations are solved by elimination or substitution. The goal is to find the point (x, y) where the two straight lines intersect.
联立方程可用消元法或代入法求解。目标是找到两条直线相交的点 (x, y)。
- Elimination method: make the coefficients of one variable match, then add or subtract the equations.
- 消元法：使其中一个变量的系数相等，然后将两个方程相加或相减。
- Substitution method: rearrange one equation to express a variable in terms of the other, then substitute.
- 代入法：将其中一个方程变形，用一个变量表示另一个变量，然后代入另一个方程。
Example: Solve 2x + y = 7 and x - y = 2.
例如：解方程组 2x + y = 7 和 x - y = 2。
- y) = 7 + 2, giving 3x = 9, so x = 3. Then substitute x = 3 into x - y = 2 to get y = 1.
将两个方程相加：(2x + y) + (x - y) = 7 + 2，得到 3x = 9，所以 x = 3。再将 x = 3 代入 x - y = 2，得到 y = 1。
- (3x - 6y) = 10 - 3, so 7y = 7 and y = 1. Substitute y = 1 into x - 2y = 1 to find x = 3.
如果系数不相等，先对一个或两个方程乘以适当的数。例如，解 3x + y = 10 和 x - 2y = 1 时，将第二个方程乘以 3，得到 3x - 6y = 3，再从第一个方程中减去它：(3x + y) - (3x - 6y) = 10 - 3，所以 7y = 7，y = 1。将 y = 1 代入 x - 2y = 1，得到 x = 3。
A simultaneous equation has exactly one solution if the two lines have different gradients. If the lines are parallel, there is no solution; if the lines coincide, there are infinitely many solutions.
如果两条直线斜率不同，联立方程恰有一个解。如果两直线平行，则无解；如果两直线重合，则有无穷多个解。
3. Solving Quadratic Equations by Factorising | 因式分解法解二次方程
+ c = 0, where a ≠ 0. If the expression can be factorised, set each factor equal to zero.
二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。如果能对表达式进行因式分解，则令每个因式等于零。
Example: Solve x² - 5x + 6 = 0.
例如：解方程 x² - 5x + 6 = 0。
se: x² - 5x + 6 = (x - 2)(x - 3). Therefore x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
因式分解：x² - 5x + 6 = (x - 2)(x - 3)。因此 x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。
Always expand the factors to check the middle term. For x² + bx + c, the two numbers must multiply to c and add to b.
一定要
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