📚 Year 9 Higher Maths: Compressed Answers and Methods | Year 9 高阶数学：压缩答案与方法速查
This revision note condenses the most common Year 9 Higher tier answers into a quick-check format. Each section gives a worked example, the final answer, and a short method note. Use it after attempting questions yourself, so that the answers become a checking tool rather than a shortcut.
本复习笔记将 Year 9 高阶数学卷中最常见的答案压缩为速查格式。每个小节提供一道例题、最终答案和简短的方法提示。请先自己尝试题目，再使用这些答案进行检查，而不是直接抄捷径。
1. Simplifying Expressions | 化简代数式
Worked example: Simplify 5x + 3y - 2x + 8y. Answer: 3x + 11y.
例题：化简 5x + 3y - 2x + 8y。答案：3x + 11y。
and 3y + 8y = 11y. Keep the x terms and y terms separate because they are not like terms.
方法：合并同类项。5x - 2x = 3x，3y + 8y = 11y。x 项和 y 项不是同类项，必须分开合并。
Common error: Adding x and y terms together to make xy or a single number. Always group only identical variable parts.
常见错误：把 x 项和 y 项相加变成 xy 或一个数。必须只对完全相同的变量部分进行合并。
2. Solving Linear Equations | 解一元一次方程
x + 19. Subtract 2x from both sides: 2x + 7 = 19. Subtract 7: 2x = 12. Divide by 2: x = 6.
例题：解方程 4x + 7 = 2x + 19。两边同时减去 2x：2x + 7 = 19。再减去 7：2x = 12。除以 2：x = 6。
x = 6
into the original equation. Left: 4(6) + 7 = 31. Right: 2(6) + 19 = 31. Both sides match.
检验：把 x = 6 代回原方程。左边：4(6) + 7 = 31。右边：2(6) + 19 = 31。两边相等。
Key rule: Do the same operation to both sides. If a term is added, subtract it. If a term is multiplied, divide by it.
关键规则：等式两边必须同时进行相同的运算。有加上的项就减去，有乘上的项就除以。
3. Expanding and Factorising | 展开与因式分解
-3x. Inner: +4x. Last: -12. Combine the x terms: -3x + 4x = x. Final answer: x² + x - 12.
例题：展开并化简 (x + 4)(x - 3)。首项：x²。外项：-3x。内项：+4x。末项：-12。合并 x 项：-3x + 4x = x。最终答案：x² + x - 12。
Factorising is the reverse process. Example: Factorise x² + 7x + 10. Find two numbers that multiply to 10 and add to 7: 2 and 5. Answer: (x + 2)(x + 5).
因式分解是展开的逆运算。例题：将 x² + 7x + 10 因式分解。找到两个数，乘积为 10，和为 7：2 和 5。答案：(x + 2)(x + 5)。
Common error: Forgetting the negative sign in the outer or inner term. Double-check the signs after expanding.
常见错误：展开时忘记外项或内项的负号。展开后要仔细检查符号。
4. Inequalities | 不等式
Worked example: Solve 5x - 3 > 12. Add 3 to both sides: 5x > 15. Divide by 5: x > 3.
例题：解不等式 5x - 3 > 12。两边同时加 3：5x > 15。两边除以 5：x > 3。
On a number line, draw an open circle at 3 and shade to the right. The open circle shows that 3 is not included.
在数轴上，在 3 处画一个空心圆，并向右画阴影。空心圆表示 3 不包含在解集内。
Important rule: If you multiply or divide both sides by a negative number, reverse the inequality sign. For example, -2x < 8 becomes x > -4.
重要规则：如果两边同时乘以或除以一个负数，必须反转不等号方向。例如，-2x < 8 变为 x > -4。
5. Linear Sequences | 线性数列
equence: when n = 1, 3n = 3, but the first term is 4. So add 1. Answer: nth term = 3n + 1.
例题：求数列 4, 7, 10, 13, 16 的第 n 项。公差是 +3，因此第 n 项从 3n 开始。将 3n 与数列比较：当 n = 1 时，3n = 3，但第一项是 4，所以要加 1。答案：第 n 项 = 3n + 1。
ch matches the second term. Substitute n = 5: 3(5) + 1 = 16, which matches the fifth term.
检验：代入 n = 2：3(2) + 1 = 7，与第二项相符。代入 n = 5：3(5) + 1 = 16，与第五项相符。
Common error: Using the first term as the multiplier instead of the common difference. The common difference gives the coefficient of n.
常见错误：用首项而不是公差作为 n 的系数。公差才是 n 的系数。
6. Ratio and Proportion | 比例与分配
Worked example: Divide £120 in the ratio 2:3. Total parts = 2 + 3 = 5. One part = £120 ÷ 5 = £24. Shares: 2 × £24 = £48 and 3 × £24 = £72.
例题：将 £120 按 2:3 的比例分配。总份数 = 2 + 3 = 5。一份 = £120 ÷ 5 = £24。分配结果：2 × £24 = £48，3 × £24 = £72。
Check: £48 + £72 = £120, so the whole amount is shared exactly.
检验：£48 + £72 = £120，因此总金额被完全分配。
For a ratio such as 1/2 : 1/4, multiply all parts by the lowest common denominator first. 1/2 : 1/4 becomes 2:1 after multiplying by 4.
对于 1/2 : 1/4 这样的比例，先将所有部分乘以最小公分母。1/2 : 1/4 乘以 4 后变成 2:1。
7. Pythagoras Theorem | 勾股定理
when the two shorter sides are 6 cm and 8 cm. Use a² + b² = c², where c is the hypotenuse.
例题：当两条较短边分别为 6 cm 和 8 cm 时，求斜边。使用 a² + b² = c²，其中 c 是斜边。
6² + 8² = c²
36 + 64 = 100
c = √100 = 10 cm
Answer: The hypotenuse is 10 cm.
答案：斜边是 10 cm。
square from the hypotenuse square. Example: hypotenuse 13 cm, one side 5 cm. Other side = √13² - 5² = √144 = 12 cm.
如果求一条较短边，用斜边平方减去已知边的平方。例题：斜边 13 cm，一条直角边 5 cm。另一条边 = √13² - 5² = √144 = 12 cm。
8. Area of 2D Shapes | 平面图形面积
parallel sides 7 cm and 13 cm, and height 4 cm. Area = ½(a + b)h = ½(7 + 13) × 4 = 40 cm².
例题：求梯形面积，平行边分别为 7 cm 和 13 cm，高为 4 cm。面积 = ½(a + b)h = ½(7 + 13) × 4 = 40 cm²。
e: Area = ½ × base × height. Example: base 10 cm, height 6 cm. Area = ½ × 10 × 6 = 30 cm².
三角形面积 = ½ × 底 × 高。例题：底 10 cm，高 6 cm。面积 = ½ × 10 × 6 = 30 cm²。
Common error: Using the slant side instead of the perpendicular height. Height must always be at right angles to the base.
常见错误：用斜边代替垂直高度。高必须与底边成直角。
9. Angles in Polygons | 多边形内角
les of a hexagon. Formula: (n - 2) × 180°. For a hexagon, n = 6, so (6 - 2) × 180° = 720°.
例题：求六边形的内角和。公式：(n - 2) × 180°。六边形 n = 6，因此 (6 - 2) × 180° = 720°。
r a regular hexagon, all interior angles are equal. Each interior angle = 720° ÷ 6 = 120°.
对于正六边形，所有内角相等。每个内角 = 720° ÷ 6 = 120°。
f any polygon is always 360°. For a regular hexagon, each exterior angle = 360° ÷ 6 = 60°.
外角方面：任何多边形的外角和始终为 360°。对于正六边形，每个外角 = 360° ÷ 6 = 60°。
10. Straight Line Graphs | 直线图像
rough (1, 2) and (4, 8). Gradient = (y₂ - y₁) / (x₂ - x₁) = (8 - 2) / (4 - 1) = 6 / 3 = 2.
例题：求经过 (1, 2) 和 (4, 8) 两点的直线斜率。斜率 = (y₂ - y₁) / (x₂ - x₁) = (8 - 2) / (4 - 1) = 6 / 3 = 2。
Equation: y = 2x + c. Substitute (1, 2): 2 = 2(1) + c, so c = 0. Final equation: y = 2x.
方程：y = 2x + c。代入 (1, 2)：2 = 2(1) + c，所以 c = 0。最终方程：y = 2x。
e gradient and c is the y-intercept. The y-intercept is where the line crosses the y-axis.
关键形式：y = mx + c，其中 m 是斜率，c 是 y 轴截距。y 轴截距是直线与 y 轴相交的位置。
11. Probability from Tables | 表格概率
red, 5 blue, and 2 green counters. One counter is taken at random. Total = 3 + 5 + 2 = 10.
例题：一个袋子中有 3 个红色、5 个蓝色和 2 个绿色计数器。随机取出一个。总数 = 3 + 5 + 2 = 10。
(red) = 3/10. P(not blue) = (3 + 2)/10 = 5/10 = 1/2. P(blue or green) = (5 + 2)/10 = 7/10.
P(红色) = 3/10。P(不是蓝色) = (3 + 2)/10 = 5/10 = 1/2。P(蓝色或绿色) = (5 + 2)/10 = 7/10。
re equally likely, probability = number of favourable outcomes ÷ total number of outcomes.
概率始终在 0 到 1 之间。如果所有结果等可能，概率 = 有利结果数 ÷ 总结果数。
12. Averages and Range | 平均数与极差
he mean, median, and range of 4, 7, 9, 12, 15. Mean = (4 + 7 + 9 + 12 + 15) ÷ 5 = 47 ÷ 5 =
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