🏠 首页 CHAPTER 4 · 章末 SUM UP

📋 Let's Sum Up!

Algebraic Notations and Algebraic Expressions

  1. An algebraic expression involves numbers and letters that are connected with operation symbols (+, −, × and ÷) and/or brackets.
  2. In an algebraic expression, there is no equal sign.
    Examples: $2n + 1$, $3a - 20bc$, $7x^2 - \dfrac{y}{z^3}$.
OperationAlgebraic expression
Addition$a + b$
Subtraction$a - b$
Multiplication$a \times b$ or $ab$;
$a \times a \times a$ or $a^3$
Division$a \div b$ or $\dfrac{a}{b}$, where $b \neq 0$

Evaluation of Algebraic Expressions and Formulae

Algebraic Expressions

We substitute each variable in an algebraic expression with its value to evaluate the expression.

Example: Substituting $a = -5$, $b = 4$ and $c = -1$,
$\dfrac{3a - c^2}{b^2} = \dfrac{3 \times (-5) - (-1)^2}{4^2} = \dfrac{-15 - 1}{16} = -\dfrac{16}{16} = -1$

Formulae

  1. A formula is an equality relating two or more variables.
  2. The value of a variable in a formula can be found by substitution when the values of other variables are known.
    Example: area $A = \pi r^2$. Substituting $r = 3$: $A = \pi \times 3^2 = 9\pi$.

Simplification of Linear Expressions

  1. Like terms have identical variable components.
    Examples: $2xy$ and $5yx$; $-6c^2$ and $2c^2$.
  2. Unlike terms do not have identical variable components.
    Examples: $2a$ and $8b$; $3x^2y$ and $xy^2$.
  3. Algebraic expressions are simplified by removing brackets and collecting like terms.
    Examples:
    $(3a - 4b) + (2a + 3b) = 3a - 4b + 2a + 3b = 3a + 2a - 4b + 3b = 5a - b$
    $(-2x + 3y) - (-3x + 4y) = -2x + 3y + 3x - 4y = -2x + 3x + 3y - 4y = x - y$

Expansion and Factorisation of Algebraic Expressions

  1. The expression $a(x + y)$ is expanded to $ax + ay$. So $ax + ay$ is the expanded form of $a(x + y)$.
  2. The distributive law is applicable when removing brackets in a linear expression.
    $a(x + y) = ax + ay \qquad a(x - y) = ax - ay \qquad a(x + y + z) = ax + ay + az$
    Examples:
    $4(-2x + 3y) = 4(-2x) + 4(3y) = -8x + 12y$
    $(-5)(7x - 2y) = (-5)(7x) - (-5)(2y) = -35x + 10y$
  3. Factorisation can be done by extracting the common factors.
    Example: $12ax - 15ya + 3a = 3a(4x) - 3a(5y) + 3a(1) = 3a(4x - 5y + 1)$
📝 Review Exercise 4

这是第 4 章的总复习。共 18 题，覆盖 §4.1–4.4。试着不翻书独立完成，再对答案。

  1. Q1. Simplify the following.
    (a) $5x \times 3y + 1 \times z = $ 检查
    (b) $z - 4x \times 6y \times y = $ 检查
    (c) $(x \times 5 - y \times y) \div 2z = $ 检查
    (d) $3x \div y - z - 5 \times y = $ 检查
  2. Q2. Write down an algebraic expression for each statement.
    (a) Subtract $c \times c$ from $d \times 5$. → 检查
    (b) Divide $x$ cubed by $y$ squared. → 检查
    (c) Multiply the product of $6a$ and 4 by $8b$. → 检查
    (d) Subtract twice the sum of $a$ and $b$ from $(3a - 2b)$. → 检查
  3. Q3. (a) Given the formula $E = \tfrac{1}{2} m(v^2 - u^2)$, find the value of $E$ when $m = 5$, $v = 11$ and $u = 7$. → $E = $ 检查
    (b) Given the formula $y = \dfrac{a - 3b^2}{(a + 3b)^2}$, find the value of $y$ when $a = 10$ and $b = -2$. → $y = $ 检查
  4. Q4. The capacity of a car is 5 passengers and that of a van is 8 passengers. Find the total capacity for $m$ cars and $n$ vans. → passengers 检查
  5. Q5. Ali is $p$ years old now.
    (a) Find his age 7 years ago in terms of $p$. → 检查
    (b) Find his age in $t$ years' time in terms of $p$ and $t$. → 检查
    (c) Ali's father is 3 years older than 4 times Ali's present age. Express his father's age in terms of $p$. → 检查
    (d) If $p = 8$, find his father's age. → 检查
  6. Q6. Simplify the following.
    (a) $(2p - 7q - 6r) + (3p - 4q - r) = $ 检查
    (b) $-(2x - 3y + 4) + (-3x + 6y - 1) = $ 检查
    (c) $(-3m - 8n + 2p) - (-4m + 7n - 3p) = $ 检查
    (d) $-(-2uw + 3w - u) - (9wu + 2u - 8w) = $ 检查
  7. Q7. Simplify the following.
    (a) $4(2m - 1) + 3(4m + 1) = $ 检查
    (b) $6(-2n - 3) - 5(2n + 6) = $ 检查
    (c) $-3(4x + y) + 2(5x - 8y) = $ 检查
    (d) $-7(2x - y + 9) - 4(-3x + y - 5) = $ 检查
  8. Q8. Express each of the following as a single fraction in its simplest form.
    (a) $\dfrac{3x}{4} + \dfrac{2(x - 1)}{5} = $ 检查
    (b) $\dfrac{4(2y - 1)}{7} - \dfrac{y - 3}{2} = $ 检查
    (c) $-\dfrac{3(t - 1)}{2} + \dfrac{5(t + 3)}{3} = $ 检查
    (d) $1 - \dfrac{3(z + 2)}{6} + \dfrac{4(1 - 2z)}{5} = $ 检查
  9. Q9. Factorise the following.
    (a) $14a + 21 = $ 检查
    (b) $28mx - 4x - 12nx = $ 检查
    (c) $-9bx - 15by = $ 检查
    (d) $15ax - 20ay + 10az = $ 检查
  10. Q10. (a) Expand and simplify $6(x + 2y) - 7(4x - 3y)$. → 检查
    (b) Factorise the result in (a). → 检查
    (c) When $x = -1$ and $y = 5$, find the value of the expression in (a). → 检查
  11. Q11. Evaluate the following expressions without using a calculator. You can evaluate the expression inside the brackets first or apply the distributive law. Which method do you prefer for each expression? Explain your answer.
    (a) $5(3.383 - 1.383) = $ 检查
    (b) $66\left(\dfrac{1}{6} - \dfrac{1}{11}\right) = $ 检查
    💡 思路

    (a) 括号里很容易算（$3.383 - 1.383 = 2$），所以先算括号；$5 \times 2 = 10$。
    (b) 分配律更快：$66 \times \tfrac{1}{6} - 66 \times \tfrac{1}{11} = 11 - 6 = 5$（避免通分）。

  12. Q12. In 5 years' time, Ann will be twice as old as Jane.
    (a) Let Jane's present age be $x$. Find Ann's age in 5 years' time in terms of $x$. → 检查
    (b) What was the sum of their ages 3 years ago in terms of $x$? → 检查
    💡 思路

    (a) Jane 现在 $x$ → 5 年后 $x+5$；Ann 5 年后 $= 2(x+5) = 2x + 10$。
    (b) Ann 现在 $= 2x + 10 - 5 = 2x + 5$。3 年前两人合计 $= (2x + 5 - 3) + (x - 3) = (2x+2) + (x-3) = 3x - 1$.
    修正：Ann 现在 = $2x + 5$。3 年前 Ann = $2x + 2$；Jane = $x - 3$。和 = $3x - 1$。
    注：textbook 给的答案是 $3x - 11$，看了下题目"Ann 5 年后是 Jane 5 年后的两倍"——所以可能原意是 Ann 现在 = $2x - 5$（即 5 年前 Ann = 2 × Jane当时）；这样 3 年前两人 = $(2x-5-3) + (x-3) = 3x - 11$。具体取决于英文原文的解读。

  13. Q13. An examination consists of three papers. The minimum total score required to pass the examination is $(8x + 4y)$ marks. Muthu scored $(2x - y + 10)$ marks and $(2x + 3y - 6)$ marks in the first two papers.
    (a) Find Muthu's total score in the first two papers. → 检查
    (b) How many marks did Muthu score in the third paper if he just passed the examination? → 检查
    (c) Factorise the result in (b). → 检查
  14. Q14. (a) Write two equivalent expressions, in terms of $a$ and $b$, to represent the area of rectangle $ABCD$ (length $= 2a + 3b$, width $= 4$).
    Expression 1: 检查
    Expression 2: 检查
    (b) Draw a diagram to show that each of the following pairs of expressions is equivalent:
    (i) $5(x + y) = 5x + 5y$
    (ii) $a(a + 5) = a^2 + 5a$
    💡 提示

    (i) 一个长 $(x+y)$、宽 $5$ 的矩形 — 切成两个矩形：一个 $x \times 5$，一个 $y \times 5$。
    (ii) 一个长 $(a+5)$、宽 $a$ 的矩形 — 切成 $a \times a$ 和 $a \times 5$。

  15. Q15. The numbers of marbles in two bags are $6ax$ and $12bx$.
    (a) Find the total number of marbles in the two bags. → 检查
    (b) Factorise the result in (a). → 检查
    (c) All the marbles are arranged in rows and columns to form a rectangle. If one side of the rectangle has $(2a + 4b)$ marbles, find the number of marbles on the other side. → 检查
    💡 思路 (c)

    $2a + 4b = 2(a + 2b)$。总数 / 一边 $= \dfrac{6x(a + 2b)}{2(a + 2b)} = 3x$。

  16. Q16. The density of zinc is 135 kg per m³ more than 7 times the density of water. The density of copper is 40 kg per m³ less than 9 times the density of water. Let $x$ kg per m³ be the density of water.
    (a) Express the density of zinc in terms of $x$. → kg/m³ 检查
    (b) Express the density of copper in terms of $x$. → kg/m³ 检查
    (c) A piece of brass, made up of zinc and copper, contains 0.02 m³ of zinc and 0.03 m³ of copper. Let $m$ kg be the mass of the piece of brass.
    (i) Given that mass = density × volume, show that the formula connecting $m$ and $x$ is $m = 0.41x + 1.5$. (Verify the relation in your own words.)
    (ii) If $x = 1000$, find the mass of the piece of brass. → kg 检查
    💡 思路 (c)(i) 验证

    $m = 0.02 \cdot (7x+135) + 0.03 \cdot (9x-40) = 0.14x + 2.7 + 0.27x - 1.2 = (0.14 + 0.27)x + (2.7 - 1.2) = 0.41x + 1.5$ ✓

  17. Q17. The length of a parcel is $a$ cm, its breadth is $b$ cm and its height is $h$ cm. The parcel is tied with red string as shown in the figure (string wraps around 2 ways across the top + once around). Find the total length of the string in terms of $a$, $b$ and $h$.
    Length = cm 检查
    💡 思路（典型答案 / 教材图示）

    红绳通常包装方式：绕一圈横向（包括长 + 高 + 长 + 高 = $2a + 2h$）+ 绕一圈纵向（宽 + 高 + 宽 + 高 = $2b + 2h$）+ 横向再加（另 2 条 $b$ 段）。具体看图，常见答案是 $2a + 4b + 6h$ cm。
    如果题图显示不同的捆绑方式（比如只绕一次），答案会是 $2a + 2b + 2h$ 或类似——请按图实测。

  18. Q18. (a) The sum of two numbers is 7. Given that the smaller number is $x$, write an expression to represent the larger number. → 检查
    (b) The difference between two numbers is 10. Given that the smaller number is $s$, write an expression to represent the larger number. → 检查
    (c) One number is 5 times another number. Given that the smaller number is $m$, what are the two possible expressions you can write to represent the larger number? (Hint: what if the two numbers are negative?)
    Possibility 1: 检查
    Possibility 2: 检查
    💡 思路 (c)

    若两数都正：smaller = $m$, larger = $5m$（因为 $5m > m$ 当 $m > 0$）。
    若两数都负：smaller 是更负的（值更小），比如 $-10$ vs $-2$；smaller = $m = -10$，larger = $-2 = m/5$。所以 larger = $m/5$。

💻 Coding

📊 Look at the flowchart below.

Start
↓
Input $x$
↓
$y = x \times (-3)$
↓
$z = 7 + y$
↓
Output $z$
↓
End
  1. (a) Express $z$ in terms of $x$. → $z = $ 检查
    💡 思路

    $y = -3x$，$z = 7 + y = 7 + (-3x) = 7 - 3x$。

  2. (b) When the input value of $x$ is 2, what is the output value? → 检查
    💡 思路

    $z = 7 - 3(2) = 7 - 6 = 1$.

  3. (c) Construct a flowchart which inputs $x$ and outputs $y = \dfrac{(5 - x)^2}{3}$.
    💡 参考流程

    Start → Input $x$ → $a = 5 - x$ → $b = a \times a$ (或 $a^2$) → $y = b \div 3$ → Output $y$ → End。 每一步只能做一个操作。

📔 Maths Journal

What are some of the rules you should observe when simplifying algebraic expressions involving brackets?

用自己的话写下来——可以列出"易错点 + 检查清单"。这是你的章末反思。

💡 检查清单（参考）

1. 嵌套括号：从最内层开始。
2. 分配律：括号前的数 / 字母 / 负号要分配给括号每一项。
3. 负号陷阱：$-(a-b) = -a+b$，不是 $-a-b$；很多错误源于此。
4. 合并同类项：变量部分必须完全相同（包括幂次）。
5. 分数：先通分（求 LCM），再合并。
6. 系数为 1 不写：$1x$ 写作 $x$；$-1y$ 写作 $-y$。
7. 验证：随便代一个数（如 $x=2$）验算前后是否相等。