🏠 首页 CHAPTER 4 · BASIC ALGEBRA AND ALGEBRAIC MANIPULATIONS 4.3

4.3

Expansion of Linear Expressions

💡 4.2 是合并同类项（变短）；4.3 是反方向——去括号（先变长再化简）。核心工具：分配律 $a(x+y) = ax + ay$。 口诀：括号前的数 / 负号 / 字母，要 分发给括号里每一项，一个都不能漏。

AExpanding Linear Expressions

In the previous section, we looked at how to simplify linear expressions. Here, we shall learn to expand linear expressions such as $3(x - 2)$ by removing brackets and writing the expression as a sum of terms.

We can use algebra discs to help us make sense of the expansion of linear expressions as illustrated in the following activity.

🧪 Activity 3 · Expanding via discs

Objective: To make sense of and interpret linear expressions involving expansion.

Examples:

(i) Expand $2(5x)$.

Take 2 groups of 5x. 2(5x) x x x x x x x x x x → Ungroup the discs. 10x x x x x x x x x x x

∴ $2(5x) = \mathbf{10x}$.

(ii) Expand $-3(4x)$.

RECALL: $-3(4)$ means the negative of 3 groups of 4. The "−" sign in $-3(4)$ means flipping over (changing signs) all 1 discs in the groups to −1 discs. Same idea here.

x x x x x x x x x x x x Flip — take the negative of 3 groups of 4x. −3(4x) → −x −x −x −x −x −x −x −x −x −x −x −x Ungroup. −12x

∴ $-3(4x) = -(12x) = \mathbf{-12x}$.

(iii) Expand $-2(-5x)$.

−x −x −x −x −x −x −x −x −x −x → x x x x x x x x x x

∴ $-2(-5x) = -(-10x) = \mathbf{10x}$.

(iv) Expand $2(3x - 1)$.

Take 2 groups of (3x − 1). 2(3x − 1) x x x −1 x x x −1 → Ungroup. 6x − 2 x x x x x x −1 −1

∴ $2(3x - 1) = \mathbf{6x - 2}$.

1. Expand the following expressions.

  1. (a) $3(4x) = $ 检查
  2. (b) $3(-4x) = $ 检查
  3. (c) $-3(-4x) = $ 检查
  4. (d) $-2(3x) = $ 检查
  5. (e) $2(-3x) = $ 检查
  6. (f) $-2(-3x) = $ 检查

2. Expand the following expressions.

  1. (a) $-2(3x - 1) = $ 检查
  2. (b) $-2(-3x - 1) = $ 检查
  3. (c) $3(x + 2y) = $ 检查
  4. (d) $-3(x + 2y) = $ 检查

3. Using the results in 1 and 2, expand the following expressions.

  1. (a) $a(bx) = $ 检查
  2. (b) $a(x + y) = $ 检查
💡 DISCUSS · How would you expand $3(-4x)$?

$3(-4x)$ 表示 "3 组 $-4x$" = $-12x$。而 $-3(4x)$ 表示 "$3$ 组 $4x$ 取负" = $-12x$。两者结果相同。可以验证 $3(-4x) = -3(4x) = -12x$. 这就是为什么 $a(-b) = -ab = (-a)(b)$.

From Activity 3, we observe that the process of expansion involves using the distributive law of multiplication over addition as follows:

$a(x + y) = ax + ay$

We say that $a(x + y)$ is expanded to $ax + ay$, or $ax + ay$ is the expanded form of $a(x + y)$.

📌 SPOTLIGHT: We can visualise the process of using the distributive law as shown:
    $2(3x - 1) = 2 \cdot 3x - 2 \cdot 1 = 6x - 2$

The distributive law can be generalised and applied as follows:

  • $a(x + y) = ax + ay$
    Multiplication can be distributed over addition.
  • $a(x - y) = ax + a(-y) = ax - ay$
    Multiplication can be distributed over subtraction.
  • $a(x + y + z) = ax + ay + az$
    Multiplication can be distributed over the sum of several terms.
📖 Worked Example 13

Expand each of the following expressions.

  1. (a) $2(3x - 4)$
  2. (b) $(-6)(-5x + 3y)$
  3. (c) $a(2x + 7y - 5z)$
  4. (d) $(2p + 4q - 5)(-3w)$

SOLUTION

(a) $2(3x - 4) = 2(3x) - 2(4) = \mathbf{6x - 8}$  (Distribute 2 over 3x − 4)

(b) $-6(-5x + 3y) = -6(-5x) + (-6)(3y) = \mathbf{30x - 18y}$  (Distribute −6 over −5x + 3y)

(c) $a(2x + 7y - 5z) = a(2x) + a(7y) - a(5z) = \mathbf{2ax + 7ay - 5az}$

(d) $(2p + 4q - 5)(-3w) = (2p)(-3w) + (4q)(-3w) + (-5)(-3w) = \mathbf{-6pw - 12qw + 15w}$

✍️ Try It Yourself 13
  1. (a) $5(2x + 7) = $ 检查
  2. (b) $(-9)(3x + 4y) = $ 检查
  3. (c) $-x(-2m - 3n + 1) = $ 检查
  4. (d) $(3x - 6y + 5z - 1)(-2b) = $ 检查

BExpanding and Simplifying Linear Expressions

In general, we can simplify an expression by first expanding the expression and then collecting like terms.

We can use algebra discs to help us make sense of the expansion and simplification of linear expressions, as illustrated in Activity 4 below.

🧪 Activity 4 · Expand-and-collect with discs

Objective: To make sense of and interpret linear expressions involving expansion and simplification.

(i) Example: Expand and simplify $(5y + 2) + 3(-2y + 1)$.

  • $(5y + 2) + 3(-2y + 1)$
  • $= 5y + 2 \underline{\;-6y + 3\;}$  (After removing brackets, the signs remain unchanged.)
  • $= 5y - 6y + 2 + 3$  (Collect like terms.)
  • $= \mathbf{-y + 5}$

(ii) Example: Expand and simplify $2(2x - 1) - 3(-x - 2)$.

  • $2(2x - 1) - 3(-x - 2)$
  • $= 4x - 2 \underline{\;+ 3x + 6\;}$  (After removing the brackets, the signs change.)
  • $= 4x + 3x - 2 + 6$  (Collect like terms.)
  • $= \mathbf{7x + 4}$

(iii) Example: Expand and simplify $-2(2x + y) - 4(-x + y)$.

  • $-2(2x + y) - 4(-x + y)$
  • $= -4x - 2y \underline{\;+ 4x - 4y\;}$  (Signs change after −4 is distributed.)
  • $= -4x + 4x - 2y - 4y$  (Collect like terms — 4 zero pairs of x and −x.)
  • $= \mathbf{-6y}$

Expand and simplify the following expressions.

  1. (a) $3(-2x + 1) + 2(4x - 1) = $ 检查
    💡 思路

    $= -6x + 3 + 8x - 2 = 2x + 1$.

  2. (b) $2(3x - 2) - 3(-x - 2) = $ 检查
    💡 思路

    $= 6x - 4 + 3x + 6 = 9x + 2$.

  3. (c) $-3(x - 2y) - 2(-2x + y) = $ 检查
    💡 思路

    $= -3x + 6y + 4x - 2y = x + 4y$.

From Activity 4, we observe that to simplify linear expressions involving brackets, we first expand the expression, then collect like terms before performing addition and subtraction.

When two expressions can be simplified to the same expression, they are equivalent.

Examples of equivalence:

  • (i) $-2(y - 1) - 2(2y + 1) = -2y + 2 - 4y - 2 = \mathbf{-6y}$
  • (ii) $-2(2x + y) - 4(-x + y) = \mathbf{-6y}$  (from Activity 4 (iii))

The expressions $-2(y-1) - 2(2y+1)$, $-2(2x+y) - 4(-x+y)$ and $-6y$ are all equivalent.

Which of the following expressions are equivalent? Why?
(a) $2(5x + 1) - 5(-2x + 1)$
(b) $-4(-5x) + 9$
(c) $3(4x - 5) - 4(-2x - 5)$
(d) $-10(-2x + 1) + 7$
💡 简化每个看看

(a) $= 10x + 2 + 10x - 5 = 20x - 3$
(b) $= 20x + 9$
(c) $= 12x - 15 + 8x + 20 = 20x + 5$
(d) $= 20x - 10 + 7 = 20x - 3$
→ (a) 和 (d) 等价（都化为 $20x - 3$）。

📖 Worked Example 14 · Nested brackets

Expand and simplify the following expressions.

  1. (a) $-2(3x - 5) + 4x$
  2. (b) $-4(5a - 3b) + 7(a - 2b)$
  3. (c) $2[x - 5(3 - x)]$

SOLUTION

(a) $-2(3x - 5) + 4x = -2(3x) - (-2)(5) + 4x = -6x + 10 + 4x = \mathbf{-2x + 10}$

(b) $-4(5a - 3b) + 7(a - 2b) = -20a + 12b + 7a - 14b = -20a + 7a + 12b - 14b = \mathbf{-13a - 2b}$

(c) $2[x - 5(3 - x)]$  (Simplify the innermost brackets first.)
$= 2[x - 5(3) - (-5)(x)]$
$= 2[x - 15 + 5x]$
$= 2[6x - 15]$
$= 2(6x) - 2(15)$
$= \mathbf{12x - 30}$

📌 SPOTLIGHT: Expand the innermost pair of brackets first.

📌 RECALL: $(-) \times (+) = (-)$; $(-) \times (-) = (+)$.

✍️ Try It Yourself 14
  1. (a) $-10y + (4 - 2y)(-5) = $ 检查
    💡 思路

    $= -10y + (-20 + 10y) = -10y - 20 + 10y = -20$.

  2. (b) $-3(-4a + 8b) + 10(2a + b) = $ 检查
    💡 思路

    $= 12a - 24b + 20a + 10b = 32a - 14b$.

  3. (c) $7y - 3[4 - 5(1 - y)] = $ 检查
    💡 思路

    Inner: $5(1-y) = 5 - 5y$. So $[4 - (5-5y)] = [4 - 5 + 5y] = [-1 + 5y]$.
    $7y - 3[-1 + 5y] = 7y + 3 - 15y = -8y + 3$.

CSimplifying Linear Expressions Involving Fractions

We can simplify expressions involving fractions by simply converting them to like fractions.

In the previous section, we use the idea of equivalent fractions to simplify expressions such as $\tfrac{1}{2}a + \tfrac{2}{3}a$. Now, we shall extend this idea to simplify linear expressions such as $\tfrac{3x - 4}{4} + \tfrac{2x + 5}{3}$.

📖 Worked Example 15

Express each of the following as a single fraction in its simplest form.

  1. (a) $\dfrac{3x - 4}{4} + \dfrac{2x + 5}{3}$
  2. (b) $\dfrac{2y}{3} - \dfrac{3(y - 5)}{2}$
  3. (c) $\dfrac{1 - 2z}{3} + \dfrac{3z + 1}{5} - \dfrac{4z - 3}{6}$

SOLUTION

(a) $\dfrac{3x - 4}{4} + \dfrac{2x + 5}{3} = \dfrac{3(3x - 4)}{12} + \dfrac{4(2x + 5)}{12}$  (LCM of 4 and 3 is 12.)
$= \dfrac{3(3x - 4) + 4(2x + 5)}{12}$  (Write as one fraction.)
$= \dfrac{9x - 12 + 8x + 20}{12}$  (Apply the distributive law.)
$= \mathbf{\dfrac{17x + 8}{12}}$  (Collect like terms.)

(b) $\dfrac{2y}{3} - \dfrac{3(y - 5)}{2} = \dfrac{2(2y)}{6} - \dfrac{3 \cdot 3(y - 5)}{6}$  (LCM of 2 and 3 is 6.)
$= \dfrac{4y - 9(y - 5)}{6} = \dfrac{4y - 9y + 45}{6} = \mathbf{\dfrac{-5y + 45}{6}}$

(c) $\dfrac{1 - 2z}{3} + \dfrac{3z + 1}{5} - \dfrac{4z - 3}{6} = \dfrac{10(1-2z) + 6(3z+1) - 5(4z-3)}{30}$  (LCM = 30)
$= \dfrac{10 - 20z + 18z + 6 - 20z + 15}{30} = \mathbf{\dfrac{-22z + 31}{30}}$

📌 RECALL: Recall the procedure for simplifying expressions involving fractional coefficients. We need to convert the fractions to like fractions with a common denominator.

📌 SPOTLIGHT: $\tfrac{2x}{3}$ can be written as $\tfrac{2}{3}x$.

What are the errors in the working shown? Find them.
$\dfrac{2x}{3} - \dfrac{3(x-5)}{2}$ (start)
$= \dfrac{2x}{3} - \dfrac{3x - 15}{2}$
$= \dfrac{2 \cdot 2x}{6} - \dfrac{3 \cdot 3x - 15}{6}$ ← ❌
$= \dfrac{2 \cdot 2x - (3 \cdot 3x - 15)}{6}$
$= \dfrac{4x - 9x + 15}{6}$
$= \dfrac{-5x + 15}{6}$
💡 错在哪？

$\dfrac{3x-15}{2}$ 想转换为分母 6 时，应该把 整个分子 乘以 3，即 $\dfrac{3(3x-15)}{6} = \dfrac{9x - 45}{6}$，而不是 $\dfrac{3 \cdot 3x - 15}{6}$。
正确：$\dfrac{4x - (9x - 45)}{6} = \dfrac{4x - 9x + 45}{6} = \dfrac{-5x + 45}{6}$（和 WE15(b) 一致）。

✍️ Try It Yourself 15

Express each of the following as a single fraction in its simplest form.

  1. (a) $-\dfrac{7x}{8} + \dfrac{3(x + 1)}{4} = $ 检查
    💡 思路

    LCM = 8. $-\tfrac{7x}{8} + \tfrac{2 \cdot 3(x+1)}{8} = \tfrac{-7x + 6(x+1)}{8} = \tfrac{-7x + 6x + 6}{8} = \tfrac{-x + 6}{8}$.

  2. (b) $\dfrac{2x + 1}{2} - \dfrac{x + 4}{3} = $ 检查
    💡 思路

    LCM = 6. $\tfrac{3(2x+1) - 2(x+4)}{6} = \tfrac{6x+3 - 2x - 8}{6} = \tfrac{4x - 5}{6}$.

  3. (c) $\dfrac{x - 1}{2} - \dfrac{2(3 - x)}{5} - \dfrac{3x + 2}{4} = $ 检查
    💡 思路

    LCM(2, 5, 4) = 20. 通分后分子 $= 10(x-1) - 8(3-x) - 5(3x+2)$
    $= 10x - 10 - 24 + 8x - 15x - 10$
    $= (10 + 8 - 15)x + (-10 - 24 - 10) = 3x - 44$。
    所以 $= \dfrac{3x - 44}{20}$。

PPractice Exercise 4.3

BASIC MASTERY
  1. Q1. Expand the following expressions.
    (a) $4(7b + 5c) = $ 检查
    (b) $(-3g - 4h)(2) = $ 检查
    (c) $-5(-3p + 9q) = $ 检查
    (d) $a(-5x + 3y - 8z) = $ 检查
    (e) $(4a - 8b + 12c)\left(-\tfrac{5}{2}\right) = $ 检查
    (f) $\tfrac{2}{3}(6a - 18b - 24c) = $ 检查
  2. Q2. Expand and simplify each of the following.
    (a) $-4(2x + 1) + 3(-x - 2) = $ 检查
    (b) $2(3u - 5) - 3(2u + 1) = $ 检查
    (c) $-5(3x - y) - 7(-2x + 3y) = $ 检查
    (d) $b(-5v - 4) - 2b(v + 4) = $ 检查
    (e) $3(2a - 3b - 5c) - 5(a - 3c) = $ 检查
    (f) $(x - 3y)(-2) + (x - y + 3)(6) = $ 检查
  3. Q3. Express each of the following as a single fraction in its simplest form.
    (a) $\dfrac{a}{18} + \dfrac{a}{6} - \dfrac{a}{3} = $ 检查
    (b) $\dfrac{a}{14} - \dfrac{2a}{7} + \dfrac{3a}{2} = $ 检查
    (c) $\dfrac{x}{2} + \dfrac{x - 8}{3} = $ 检查
    (d) $\dfrac{x}{5} - \dfrac{2x - 1}{3} = $ 检查
    (e) $\dfrac{x + 2}{3} + \dfrac{1 - 3x}{4} = $ 检查
    (f) $\dfrac{2x + 1}{4} - \dfrac{x - 3}{5} = $ 检查
INTERMEDIATE
  1. Q4. Expand and simplify each of the following.
    (a) $4a - [5a - (3 + 2a)] = $ 检查
    (b) $7t - [5s + 8(s + 2t)] = $ 检查
    (c) $4m + n + [5m - 6(m - n)] = $ 检查
    (d) $5[a - (b - a)] + 7(-a + 2b) = $ 检查
    (e) $9(y - z) - [5y - z - 3(2y - 4z)] = $ 检查
    (f) $4[2p + 3q - (p + q)] = $ 检查
  2. Q5. Expand and simplify each of the following.
    (a) $\tfrac{5}{2}[-7x - 3(2 - 5x)] = $ 检查
    (b) $\tfrac{1}{3}[16 - y - 7(1 + 2y)] = $ 检查
    (c) $5p - 2[4(p - 3q) - 3(-p - 2q)] = $ 检查
    (d) $3\{2r - [10s - (4r + 5s)]\} = $ 检查
  3. Q6. Express each of the following as a single fraction in its simplest form.
    (a) $-\dfrac{3t}{7} + \dfrac{t + 8}{3} = $ 检查
    (b) $-\dfrac{5n}{3} - \dfrac{2(n - 1)}{9} = $ 检查
    (c) $\dfrac{3(1 - n)}{2} + \dfrac{4n - 3}{5} = $ 检查
    (d) $\dfrac{5t - 2}{5} - \dfrac{2(t + 1)}{3} = $ 检查
    (e) $-\dfrac{5(x - 1)}{6} - \dfrac{3(2x + 1)}{4} = $ 检查
    (f) $1 - \dfrac{x + 2}{3} + \dfrac{4(3x - 1)}{9} = $ 检查
    (g) $\dfrac{y + 1}{3} + \dfrac{y + 2}{2} - \dfrac{5y}{6} = $ 检查
    (h) $\dfrac{y}{2} - \dfrac{3(1 - 3y)}{5} - \dfrac{2(4y + 7)}{3} = $ 检查
  4. Q7. There are $(2a + b)$ books in a pile. The thickness of each book is 2 cm. Find the height of the pile of books in terms of $a$ and $b$, expressing your answer in the expanded form. → cm 检查
  5. Q8. A grocer bought $n$ eggs at $x each. He marked up the price of each egg by $y and sold all the eggs. Find the total amount he collected in terms of $n$, $x$ and $y$, expressing your answer in the expanded form. → $ 检查
ADVANCED
  1. Q9. Sam has some oranges. He packs them in rows in a box. There are six rows with $(2m - 3)$ oranges each and one row with 5 oranges.
    (a) Express, in expanded form, the number of oranges in the box in terms of $m$. → 检查
    (b) Find the number of oranges in the box when $m = 7$. → 检查
    💡 思路

    (a) $6(2m-3) + 5 = 12m - 18 + 5 = 12m - 13$. (b) $12(7) - 13 = 84 - 13 = 71$.

  2. Q10. In making the frame of a rectangular box, a carpenter needs 4 wooden sticks of length $(2x + 3y)$ cm each and 8 wooden sticks of length $(2x + y)$ cm each.
    (a) Express the total length of wood required in terms of $x$ and $y$. → cm 检查
    (b) If $x = 30$ and $y = 10$, find the total length of wood required in centimetres. → cm 检查
  3. Q11. A condominium has 28 floor levels. In each of the lower 20 floor levels, there are $x$ 3-bedroom and $y$ 2-bedroom apartments. In each of the upper 8 floor levels, there are $(x - 1)$ 4-bedroom apartments.
    (a) Express the total number of bedrooms in the condominium in terms of $x$ and $y$. → 检查
    (b) If $x = 5$ and $y = 3$, find the total number of bedrooms in the condominium. → 检查
    💡 思路

    Lower 20 floors × ($3x + 2y$) bedrooms/floor = $60x + 40y$. Upper 8 floors × $(x-1) \cdot 4 = 8 \cdot 4(x-1) = 32(x-1) = 32x - 32$. 合 = $92x + 40y - 32$. $x=5, y=3$: $92(5)+40(3)-32 = 460+120-32 = 548$.

  4. Q12. The distributive law can be applied in mental calculation. For example, $37 \times 99 = 37 \times (100 - 1) = 3700 - 37 = 3663$. Using the distributive law, find the value of
    (a) $98 \times 30 = $ 检查
    (b) $25 \times 102 = $ 检查
    (c) $99 \times 99 = $ 检查
    💡 思路

    (a) $(100-2)(30) = 3000 - 60 = 2940$.
    (b) $25(100+2) = 2500 + 50 = 2550$.
    (c) $99(100-1) = 9900 - 99 = 9801$.

  5. Q13. (a) Simplify the given expression $2\left[\dfrac{m(x - 4)}{3} - 1\right] + \dfrac{m}{5}$ when (i) $x = 5$, (ii) $x = 4$.
    (i) Value when $x = 5$: 检查
    (ii) Value when $x = 4$: 检查
    (b) Given that the expression in (a) has the same value for all values of $x$, find
    (i) the value of $m$: 检查
    (ii) the value of the expression: 检查
    💡 思路

    展开：$2 \cdot \tfrac{m(x-4)}{3} - 2 + \tfrac{m}{5} = \tfrac{2m(x-4)}{3} - 2 + \tfrac{m}{5}$。
    要让结果与 $x$ 无关，必须让 $x$ 项的系数 $= 0$，即 $\tfrac{2m}{3} = 0 \Rightarrow m = 0$。
    代回 $m = 0$：表达式 $= 0 - 2 + 0 = -2$.