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Algebraic Techniques, SM-Bank 121

Use the distributive law to expand the expression $x(x+2)$. (2 marks)

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$x^2+2x$

Show Worked Solution

$x(x+2)$	$=x\times x+x\times 2$
	$=x^2+2x$

Algebraic Techniques, SM-Bank 120

Use the distributive law to expand the expression $-(a-8)$. (2 marks)

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$-a+8$

Show Worked Solution

$-(a-8)$	$=-1\times a-(-1)\times 8$
	$=-a+8$

Algebraic Techniques, SM-Bank 119

Use the distributive law to expand the expression $-10(y+4)$. (2 marks)

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$-10y-40$

Show Worked Solution

$-10(y+4)$	$=-10\times y+(-10)\times 4$
	$=-10y-40$

Algebraic Techniques, SM-Bank 118

Use the distributive law to expand the expression $4(m-3)$. (2 marks)

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$4m-12$

Show Worked Solution

$4(m-3)$	$=4\times m-4\times 3$
	$=4m-12$

Algebraic Techniques, SM-Bank 117

Use the distributive law to expand the expression $3(x+1)$. (2 marks)

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$3x+3$

Show Worked Solution

$3(x+1)$	$=3\times x+3\times 1$
	$=3x+3$

Algebraic Techniques, SM-Bank 116

Use the algebraic expression $4-x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$	$3$	$2$	$1$	$0$	$-1$	$-6$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$4-x$	$3$	$2$	$1$	$0$	$-1$	$-6$

Algebraic Techniques, SM-Bank 115

Use the algebraic expression $-x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$	$-1$	$-2$	$-3$	$-4$	$-5$	$-10$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$-x$	$-1$	$-2$	$-3$	$-4$	$-5$	$-10$

Algebraic Techniques, SM-Bank 114

Use the algebraic expression $5x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$	$5$	$10$	$15$	$20$	$25$	$50$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$5x$	$5$	$10$	$15$	$20$	$25$	$50$

Algebraic Techniques, SM-Bank 113

Use the algebraic expression $x^2-3x$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$	$-2$	$-2$	$0$	$4$	$10$	$70$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2-3x$	$-2$	$-2$	$0$	$4$	$10$	$70$

Algebraic Techniques, SM-Bank 112

Use the algebraic expression $x^2+1$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$	$2$	$5$	$10$	$17$	$26$	$101$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x^2+1$	$2$	$5$	$10$	$17$	$26$	$101$

Algebraic Techniques, SM-Bank 111

Use the algebraic expression $2x+1$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$	$3$	$5$	$7$	$9$	$11$	$21$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$2x+1$	$3$	$5$	$7$	$9$	$11$	$21$

Algebraic Techniques, SM-Bank 110

Use the algebraic expression $x-5$ to complete the table. (3 marks)

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$

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$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$	$-4$	$-3$	$-2$	$-1$	$0$	$5$

Show Worked Solution

$x$	$1$	$2$	$3$	$4$	$5$	$10$
$x-5$	$-4$	$-3$	$-2$	$-1$	$0$	$5$

Algebraic Techniques, SM-Bank 109

Use the algebraic expression $x+3$ to complete the table. (3 marks)

$x$	1	2	3	4	5	10
$x+3$

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$x$	1	2	3	4	5	10
$x+3$	4	5	6	7	8	13

Show Worked Solution

$x$	1	2	3	4	5	10
$x+3$	4	5	6	7	8	13

Algebraic Techniques, SM-Bank 108

Simplify $\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}$ giving your answer as an algebraic fraction in simplest form. (2 marks)

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$\dfrac{3x}{5}$

Show Worked Solution

$\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}$	$=\dfrac{3x}{2}\times \dfrac{8}{5x}\times \dfrac{x}{4}$
	$=\dfrac{3x}{5}$

Algebraic Techniques, SM-Bank 107

Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.

$\dfrac{5}{y}\ ÷\ \dfrac{4}{x}$ (1 mark)

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$\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}$ (1 mark)

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$\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}$ (2 marks)

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a. $\dfrac{5x}{4y}$

b. $\dfrac{9ab}{7c}$

c. $\dfrac{4rs}{45}$

Show Worked Solution

a.	$\dfrac{5}{y}\ ÷\ \dfrac{4}{x}$	$=\dfrac{5}{y}\times \dfrac{x}{4}$
		$=\dfrac{5x}{4y}$

b.	$\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}$	$=\dfrac{9a}{7}\times\dfrac{b}{c}$
		$=\dfrac{9ab}{7c}$

c.	$\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}$	$=\dfrac{r}{3}\times \dfrac{4s}{5}$
		$=\dfrac{4rs}{15}$

Algebraic Techniques, SM-Bank 106

Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.

$\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}$ (2 marks)

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$\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}$ (2 marks)

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a. $\dfrac{5}{2}$

b. $\dfrac{8x}{y}$

Show Worked Solution

a.	$\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}$	$=\dfrac{5m}{8}\times \dfrac{4}{m}$
		$=\dfrac{20m}{8m}$
		$=\dfrac{5}{2}$

b.	$\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}$	$=\dfrac{14x}{3}\times \dfrac{6}{7y}$
		$=\dfrac{7x\times 2\times 3\times 2}{7y\times 3}$
		$=\dfrac{4x}{y}$

Algebraic Techniques, SM-Bank 105

Simplify the following products, giving your answer as an algebraic fraction in simplest form.

$\dfrac{2b}{3}\times \dfrac{9c}{5}$ (2 marks)

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$\dfrac{8x}{5}\times \dfrac{25y}{12}$ (2 marks)

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a. $\dfrac{6bc}{5}$

b. $\dfrac{10xy}{3}$

Show Worked Solution

a.	$\dfrac{2b}{3}\times \dfrac{9c}{5}$	$=\dfrac{2b\times 9c}{3\times 5}$
		$=\dfrac{18bc\ ÷\ 3}{15\ ÷\ 3}$
		$=\dfrac{6bc}{5}$

b.	$\dfrac{8x}{5}\times \dfrac{25y}{12}$	$=\dfrac{8x\times 25y}{5\times 12}$
		$=\dfrac{200xy\ ÷\ 20}{60\ ÷\ 20}$
		$=\dfrac{10xy}{3}$

Algebraic Techniques, SM-Bank 104

Simplify the following products, giving your answer as an algebraic fraction in simplest form.

$\dfrac{a}{2}\times \dfrac{1}{5}$ (1 mark)

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$\dfrac{3x}{5}\times \dfrac{2y}{7}$ (1 mark)

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$\dfrac{5r}{9}\times \dfrac{4s}{5}$ (2 marks)

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a. $\dfrac{a}{10}$

b. $\dfrac{6xy}{35}$

c. $\dfrac{4rs}{9}$

Show Worked Solution

$\dfrac{a}{2}\times \dfrac{1}{5}$

$=\dfrac{a\times 1}{2\times 5}=\dfrac{a}{10}$

$\dfrac{3x}{5}\times \dfrac{2y}{7}$

$=\dfrac{3x\times 2y}{5\times 7}=\dfrac{6xy}{35}$

$\dfrac{5r}{9}\times \dfrac{4s}{5}$

$=\dfrac{5r\times 4s}{9\times 5}=\dfrac{4rs}{9}$

Algebraic Techniques, SM-Bank 103

For the expression $\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{11a}{12}$

Show Worked Solution

$\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}$	$=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{2a}{3}\times \dfrac{4}{4}-\dfrac{a}{4}\times \dfrac{3}{3}$
	$=\dfrac{6a}{12}+\dfrac{8a}{12}-\dfrac{3a}{12}$
	$=\dfrac{11a}{12}$

Algebraic Techniques, SM-Bank 102

Ben recieved $$x$ for his birthday.

He spent $\dfrac{1}{2}$ of his money on shoes and $\dfrac{1}{3}$ on Gold Class movie tickets.

Write an algebraic fraction to represent the amount he spent on shoes. (1 mark)

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Write an algebraic fraction to represent the amount he spent on movie tickets. (1 mark)

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Using part (a) and (b) above, write a simplified algebraic fraction to represent the total amount of money he has spent so far. (2 marks)

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a. $\dfrac{$x}{2}$

b. $\dfrac{$x}{3}$

c. $\dfrac{$5x}{6}$

Show Worked Solution

a.	$\text{Shoes}$	$=\dfrac{1}{2}\times x$
		$=\dfrac{$x}{2}$

b.	$\text{Movie tickets}$	$=\dfrac{1}{3}\times x$
		$=\dfrac{$x}{3}$

c.	$\text{Total Spent}$	$=\dfrac{x}{2}+\dfrac{x}{3}$
		$=\dfrac{x}{2}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{2}{2}$
		$=\dfrac{3x}{6}+\dfrac{2x}{6}$
		$=\dfrac{$5x}{6}$

Algebraic Techniques, SM-Bank 101

Yesterday Gareth walked $x$ kilometres in Kosciuszko National park.

He started at Thredbo village and walked up Merritts Nature Track to the Kosciuszko express chairlift. When he arrived at the chairlift he had complete $\dfrac{1}{5}$ of the total distance.

Gareth then joined a walking group and walked a further $\dfrac{1}{3}$ of the total distance before beginning his descent back to the village.

Write an algebraic fraction to represent the distance he walked to the Kosciuszko express chairlift. (1 mark)

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Write an algebraic fraction to represent the distance he walked with the walking group. (1 mark)

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Using part (a) and (b) above, write a simplified algebraic fraction to represent the total distance Gareth covered in the first two legs of his walk. (2 marks)

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a. $\dfrac{x}{5}\ \text{kilometres}$

b. $\dfrac{x}{3}\ \text{kilometres}$

c. $\dfrac{8x}{15}\ \text{kilometres}$

Show Worked Solution

a.	$\text{First leg}$	$=\dfrac{1}{5}\times x$
		$=\dfrac{x}{5}\ \text{kilometres}$

b.	$\text{Second leg}$	$=\dfrac{1}{3}\times x$
		$=\dfrac{x}{3}\ \text{kilometres}$

c.	$\text{Distance travelled}$	$=\dfrac{x}{5}+\dfrac{x}{3}$
		$=\dfrac{x}{5}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{5}{5}$
		$=\dfrac{3x}{15}+\dfrac{5x}{15}$
		$=\dfrac{8x}{15}\ \text{kilometres}$

Algebraic Techniques, SM-Bank 100

For the sum $a+\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5a}{4}$

Show Worked Solution

$a+\dfrac{a}{4}$	$=\dfrac{a}{1}\times \dfrac{4}{4}+\dfrac{a}{4}$
	$=\dfrac{4a}{4}+\dfrac{a}{4}$
	$=\dfrac{5a}{4}$

Algebraic Techniques, SM-Bank 099

For the difference $\dfrac{a}{3}-\dfrac{b}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{4a-3b}{12}$

Show Worked Solution

$\dfrac{a}{3}-\dfrac{b}{4}$	$=\dfrac{a}{3}\times \dfrac{4}{4}-\dfrac{b}{4}\times \dfrac{3}{3}$
	$=\dfrac{4a}{12}-\dfrac{3b}{12}$
	$=\dfrac{4a-3b}{12}$

Algebraic Techniques, SM-Bank 098

For the difference $\dfrac{5r}{8}-\dfrac{3r}{8}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{r}{4}$

Show Worked Solution

$\dfrac{5r}{8}-\dfrac{3r}{8}$	$=\dfrac{2r}{8}$
	$=\dfrac{r}{4}$

Algebraic Techniques, SM-Bank 097

For the difference $\dfrac{m}{2}-\dfrac{m}{7}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5m}{14}$

Show Worked Solution

$\dfrac{m}{2}-\dfrac{m}{7}$	$=\dfrac{m}{2}\times \dfrac{7}{7}-\dfrac{m}{7}\times \dfrac{2}{2}$
	$=\dfrac{7m}{14}-\dfrac{2m}{14}$
	$=\dfrac{5m}{14}$

Algebraic Techniques, SM-Bank 096

For the difference $\dfrac{a}{4}-\dfrac{a}{5}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{a}{20}$

Show Worked Solution

$\dfrac{a}{4}-\dfrac{a}{5}$	$=\dfrac{a}{4}\times \dfrac{5}{5}-\dfrac{a}{5}\times \dfrac{4}{4}$
	$=\dfrac{5a}{20}-\dfrac{4a}{20}$
	$=\dfrac{a}{20}$

Algebraic Techniques, SM-Bank 095

For the difference $\dfrac{2x}{3}-\dfrac{x}{4}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5x}{12}$

Show Worked Solution

$\dfrac{2x}{3}-\dfrac{x}{4}$	$=\dfrac{2x}{3}\times \dfrac{4}{4}-\dfrac{x}{4}\times \dfrac{3}{3}$
	$=\dfrac{8x}{12}-\dfrac{3x}{12}$
	$=\dfrac{5x}{12}$

Algebraic Techniques, SM-Bank 094

For the sum $\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}$, simplify and write an equivalent algebraic fraction. (3 marks)

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$\dfrac{13a}{12}$

Show Worked Solution

$\dfrac{a}{2}+\dfrac{a}{3}+\dfrac{a}{4}$	$=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{a}{3}\times \dfrac{4}{4}+\dfrac{a}{4}\times \dfrac{3}{3}$
	$=\dfrac{6a}{12}+\dfrac{4a}{12}+\dfrac{3a}{12}$
	$=\dfrac{13a}{12}$

Algebraic Techniques, SM-Bank 093

For the sum $\dfrac{b}{3}+\dfrac{5b}{6}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{7b}{6}$

Show Worked Solution

$\dfrac{b}{3}+\dfrac{5b}{6}$	$=\dfrac{b}{3}\times \dfrac{2}{2}+\dfrac{5b}{6}$
	$=\dfrac{2b}{6}+\dfrac{5b}{6}$
	$=\dfrac{7b}{6}$

Algebraic Techniques, SM-Bank 092

For the sum $\dfrac{2x}{3}+\dfrac{x}{5}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{13x}{15}$

Show Worked Solution

$\dfrac{2x}{3}+\dfrac{x}{5}$	$=\dfrac{2x}{3}\times \dfrac{5}{5}+\dfrac{x}{5}\times \dfrac{3}{3}$
	$=\dfrac{10x}{15}+\dfrac{3x}{15}$
	$=\dfrac{13x}{15}$

Algebraic Techniques, SM-Bank 091

For the sum $\dfrac{a}{4}+\dfrac{a}{2}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{3a}{4}$

Show Worked Solution

$\dfrac{a}{4}+\dfrac{a}{2}$	$=\dfrac{a}{4}+\dfrac{a}{2}\times \dfrac{2}{2}$
	$=\dfrac{a}{4}+\dfrac{2a}{4}$
	$=\dfrac{3a}{4}$

Algebraic Techniques, SM-Bank 090

For the sum $\dfrac{m}{3}+\dfrac{m}{2}$, simplify and write an equivalent algebraic fraction. (2 marks)

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$\dfrac{5m}{6}$

Show Worked Solution

$\dfrac{m}{3}+\dfrac{m}{2}$	$=\dfrac{m}{3}\times \dfrac{2}{2}+\dfrac{m}{2}\times \dfrac{3}{3}$
	$=\dfrac{2m}{6}+\dfrac{3m}{6}$
	$=\dfrac{5m}{6}$

Algebraic Techniques, SM-Bank 089 MC

$\dfrac{2}{m}$ is equivalent to:

$\dfrac{4}{8m}$
$\dfrac{3m}{12}$
$\dfrac{6m}{3m^2}$
$\dfrac{1}{m}+\dfrac{1}{m}$

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$C$

Show Worked Solution

$\text{Consider Option C:}$

$\dfrac{6m}{3m^2}$	$=\dfrac{3\times 2\times m}{3\times m\times m}$
	$=\dfrac{2}{m}$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 088 MC

$\dfrac{a}{4}$ is equivalent to:

$\dfrac{4a}{a}$
$\dfrac{3a}{12}$
$\dfrac{a^2}{4a^3}$
$\dfrac{a^4}{a^3}$

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$B$

Show Worked Solution

$\text{Consider Option B:}$

$\dfrac{3a}{12}$	$=\dfrac{3\times a}{3\times 4}$
	$=\dfrac{a}{4}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 087 MC

$\dfrac{x}{3}$ is equivalent to:

$\dfrac{2x}{6}$
$x+\dfrac{1}{3}$
$\dfrac{12x}{4}$
$\dfrac{x^3}{x^2}$

Show Answers Only

$A$

Show Worked Solution

$\text{Consider Option A:}$

$\dfrac{2x}{6}$	$=\dfrac{2\times x}{2\times 3}$
	$=\dfrac{x}{3}$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 086

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{5xy\times 3xy}{10\times xy}$

Show Answers Only

$\dfrac{3xy}{2}$

Show Worked Solution

$\dfrac{5xy\times 3xy}{10\times xy}$	$=\dfrac{15\times xy\times xy}{10\times xy}$
	$=\dfrac{15xy}{10}$
	$=\dfrac{3xy}{2}$

Algebraic Techniques, SM-Bank 085

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{3\times ab\times 4a}{a\times 2b}$

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$6a$

Show Worked Solution

$\dfrac{3\times ab\times 4a}{a\times 2b}$	$=\dfrac{12\times a\times ab}{2\times ab}$
	$=6a$

Algebraic Techniques, SM-Bank 084

Simplify the following algebraic expression, giving your answer in simplest form. (2 marks)

$\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}$

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$n$

Show Worked Solution

$\dfrac{4m\times 3mn\times 5n}{30m\times 2mn}$	$=\dfrac{60\times m\times mn\times n}{60\times m\times mn}$
	$=n$

Algebraic Techniques, SM-Bank 083

A rectangular prism is known to have a length of $5$ cm, a width of $3$ cm and a height of $x+2$ cm. Write an expression for the volume of the prism, giving your answer in simplest form. (3 marks)

Show Answers Only

$(15x+30)\ \text{cm}^3$

Show Worked Solution

$\text{Volume}$	$=\text{length}\times\text{width}\times\text{height}$
	$=5\times 3\times (x+2)$
	$=15\times (x+2)$
	$=15\times x +15\times 2$
	$=(15x+30)\ \text{cm}^3$

Algebraic Techniques, SM-Bank 082

Write a simplified expression for the volume of the rectangular prism below. (3 marks)

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$24abc$

Show Worked Solution

$\text{Volume}$	$=\text{length}\times\text{width}\times\text{height}$
	$=4c\times 3a\times 2b$
	$=24abc$

Algebraic Techniques, SM-Bank 081

Write a simplified expression for the volume of the cube below. (3 marks)

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$8x^3$

Show Worked Solution

$\text{Volume}$	$=\text{(side)}^3$
	$=(2x)^3$
	$=2x\times 2x\times 2x$
	$=8x^3$

Algebraic Techniques, SM-Bank 080

Write a simplified expression for the area of the shape below. (3 marks)

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$48ab$

Show Worked Solution

$\text{Area}$	$=\text{Area big rectangle}-\text{Area small rectangle}$
	$=6a\times 10b-2a\times 6b$
	$=60ab-12ab$
	$=48ab$

Algebraic Techniques, SM-Bank 079

Write a simplified expression for the area of the rectangle below. (2 marks)

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$28b^2$

Show Worked Solution

$\text{Area}$	$=\text{length}\times \text{width}$
	$=7b\times 4b$
	$=28b^2$

Algebraic Techniques, SM-Bank 078

Write a simplified expression for the area of the rectangle below. (2 marks)

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$10mn$

Show Worked Solution

$\text{Area}$	$=\text{length}\times \text{width}$
	$=5m\times 2n$
	$=10mn$

Algebraic Techniques, SM-Bank 077

Write a simplified expression for the area of the triangle below. (2 marks)

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$3x$

Show Worked Solution

$\text{Area}$	$=\dfrac{1}{2}\times\text{base}\times \text{height}$
	$=\dfrac{1}{2}\times 2x\times 3$
	$=\dfrac{1}{2}\times 6x$
	$=3x$

Algebraic Techniques, SM-Bank 076

The side length of a regular hexagon is $2x+5$. Write an expression for the perimeter of the hexagon. (2 marks)

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$6\times(2x+5)\ \ \text{or }\ 6(2x+5)\ \ \text{or }\ 12x+30$

Show Worked Solution

$\text{A regular hexagon has all sides equal}$

$\therefore\ \text{Perimeter}$	$=6\times\text{side length}$
	$=6\times(2x+5)$
	$=6\times 2x+6\times 5$
	$=12x+30$

Algebraic Techniques, SM-Bank 075

The side length of a square is $x-3$. Write an expression for the perimeter of the square. (2 marks)

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$4\times(x-3)\ \ \text{or }\ 4(x-3)\ \ \text{or }\ 4x-12$

Show Worked Solution

$\text{A square has all sides equal}$

$\therefore\ \text{Perimeter}$	$=4\times\text{side length}$
	$=4\times(x-3)$
	$=4\times x-4\times 3$
	$=4x-12$

Algebraic Techniques, SM-Bank 074

The side length of an equilateral triangle is $x+2$. Write an expression for the perimeter of the triangle. (2 marks)

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$3\times(x+2)\ \ \text{or }\ 3(x+2)\ \ \text{or }\ 3x+6$

Show Worked Solution

$\text{An equilateral triangle has all sides equal}$

$\therefore\ \text{Perimeter}$	$=3\times\text{side length}$
	$=3\times(x+2)$
	$=3\times x+3\times 2$
	$=3x+6$

Algebraic Techniques, SM-Bank 073

The perimeter of a regular hexagon is equal to $4x+3y$. Write an expression for the side length of the hexagon. (2 marks)

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$\dfrac{4x+3y}{6}$

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$\text{A regular hexagon has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{6}$
	$=\dfrac{4x+3y}{6}$

Algebraic Techniques, SM-Bank 072

The perimeter of a square is equal to $5q-3$. Write an expression for the side length of the square. (2 marks)

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$\dfrac{5q-3}{4}$

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$\text{A square has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{4}$
	$=\dfrac{5q-3}{4}$

Algebraic Techniques, SM-Bank 071

The perimeter of an equilateral triangle is equal to $4m+5$. Write an expression for the side length of the triangle. (2 marks)

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$\dfrac{4m+5}{3}$

Show Worked Solution

$\text{An equilateral triangle has all sides equal}$

$\therefore\ \text{Side length}$	$=\dfrac{\text{Perimeter}}{3}$
	$=\dfrac{4m+5}{3}$

Algebraic Techniques, SM-Bank 070 MC

Which of the following expressions is equivalent to $(m\times 4)÷(3\times n)$?

$\dfrac{3m}{4n}$
$\dfrac{3n}{4m}$
$\dfrac{3mn}{4}$
$\dfrac{4m}{3n}$

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$D$

Show Worked Solution

$(m\times 4)÷(3\times n)$

$=\dfrac{m\times 4}{3\times n}$

$=\dfrac{4m}{3n}$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 069 MC

Which of the following expressions is equivalent to $m÷(3\times n)$?

$\dfrac{m}{3n}$
$\dfrac{3}{mn}$
$\dfrac{3m}{n}$
$\dfrac{mn}{3}$

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$A$

Show Worked Solution

$m÷(3\times n)$

$=\dfrac{m}{3\times n}$

$=\dfrac{m}{3n}$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 068 MC

Which of the following expressions is equivalent to $x÷2\times y$?

$\dfrac{2x}{y}$
$\dfrac{xy}{2}$
$\dfrac{2y}{x}$
$\dfrac{x}{2y}$

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$B$

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$x÷2\times y$

$=\dfrac{x}{2}\times y$

$=\dfrac{xy}{2}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 067 MC

Which of the following expressions is equivalent to $a÷b\times c$?

$\dfrac{ac}{b}$
$\dfrac{a}{bc}$
$\dfrac{ab}{c}$
$\dfrac{bc}{a}$

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$A$

Show Worked Solution

$a÷b\times c$

$=\dfrac{a}{b}\times c$

$=\dfrac{ac}{b}$

$\Rightarrow A$

Algebraic Techniques, SM-Bank 066 MC

Which of the following expressions is equivalent to $a÷(b\times c)$?

$\dfrac{ac}{b}$
$\dfrac{a}{bc}$
$\dfrac{ab}{c}$
$\dfrac{bc}{a}$

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$B$

Show Worked Solution

$a÷(b\times c)$

$=\dfrac{a}{b\times c}$

$=\dfrac{a}{bc}$

$\Rightarrow B$

Algebraic Techniques, SM-Bank 065

Simplify the following algebraic expressions, giving your answer in simplest form.

$\dfrac{9mn}{3}$ (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
$\dfrac{4y}{12x}$ (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
$\dfrac{16a^2}{-4ab}$ (2 marks)

--- 1 WORK AREA LINES (style=lined) ---
$\dfrac{-18b^2}{-24b}$ (2 marks)

--- 1 WORK AREA LINES (style=lined) ---

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a. $3mn$

b. $\dfrac{y}{3x}$

c. $\dfrac{-4a}{b}$

d. $\dfrac{3b}{4}$

Show Worked Solution

a.	$\dfrac{9mn}{3}$	$=\dfrac{3\times 3\times m\times n}{3}$
		$=3mn$

b.	$\dfrac{4y}{12x}$	$=\dfrac{4\times y}{4\times 3\times x} $
		$=\dfrac{y}{3x}$

c.	$\dfrac{16a^2}{-4ab}$	$=\dfrac{4\times 4\times a\times a}{-4\times a\times b}$
		$=\dfrac{-4a}{b}$

d.	$\dfrac{-18b^2}{-24b}$	$=\dfrac{-6\times 3\times b\times b}{-6\times 4\times b}$
		$=\dfrac{3b}{4}$

Algebraic Techniques, SM-Bank 064 MC

Which of the following algebraic expressions is equivalent to $\dfrac{20xy}{5yz}$?

$5xy^2z$
$4xy^2z$
$\dfrac{4}{z}$
$\dfrac{4x}{z}$

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$D$

Show Worked Solution

$\dfrac{20xy}{5yz}$

$=\dfrac{5\times 4\times x\times y}{5\times y\times z}$

$=\dfrac{4x}{z}$

$\Rightarrow D$

Algebraic Techniques, SM-Bank 063 MC

Which of the following algebraic expressions is equivalent to $x\times x\times y\times x\times y$?

$6xy$
$x^2y^3$
$x^3y^2$
$3x+2y$

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$C$

Show Worked Solution

$x\times x\times y\times x\times y$

$=x\times x\times x\times y\times y$

$=x^3y^2$

$\Rightarrow C$

Algebraic Techniques, SM-Bank 062 MC

Which of the following algebraic expressions is equivalent to $3a\times 2b\times ab\times 2$?

$7ab$
$12ab$
$6a^2b^2$
$12a^2b^2$

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$D$

Show Worked Solution

$3a\times 2b\times ab\times 2$

$=3\times 2\times 2\times a\times a\times b\times b$

$=12a^2b^2$

$\Rightarrow D$

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(10\)
\(4-x\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(10\)
\(4-x\)	\(3\)	\(2\)	\(1\)	\(0\)	\(-1\)	\(-6\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(10\)
\(4-x\)	\(3\)	\(2\)	\(1\)	\(0\)	\(-1\)	\(-6\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(10\)
\(-x\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(10\)
\(-x\)	\(-1\)	\(-2\)	\(-3\)	\(-4\)	\(-5\)	\(-10\)

\(x(x+2)\)	\(=x\times x+x\times 2\)
	\(=x^2+2x\)

\(-(a-8)\)	\(=-1\times a-(-1)\times 8\)
	\(=-a+8\)

\(-10(y+4)\)	\(=-10\times y+(-10)\times 4\)
	\(=-10y-40\)

\(4(m-3)\)	\(=4\times m-4\times 3\)
	\(=4m-12\)