Introductory algebra The Body Mass Index is used as an indicator of whether or not people are in a healthy weight range for their height.

Transcription

1 Introductory algebra The Body Mass Index is used as an indicator of whether or not people are in a healthy weight range for their height. If B represents the Body Mass Index, m is the person s mass in kilograms and h is the person s height in metres, m then B = h A person is considered to be in a healthy weight range if B 5. Can you comment on Richard s weight given that he is.75 m tall and has a mass of 86 kg? Completing this chapter will refresh your skills in using pronumerals to represent unknown numbers and quantities. You will learn how to simplify algebraic expressions and substitute known values into formulas.

2 80 Maths Quest 9 for Victoria Using pronumerals The language of algebra Like English, French or a computer language, algebra is a type of language. When reading a language we learn to recognise the various written parts. Consider the following algebraic sentence. 4xy + 5x y = 7xy + y This can be broken down as follows: Algebra object Name How do we recognise it? 4 x y 4xy + 5x y 7xy y 4xy + 5x y 7xy + y Coefficient Pronumeral Pronumeral Term Term Term Term Term Constant term Expression Expression 4xy + 5x y = 7xy + y Equation The number part of a term A letter part of the term A letter part of the term A group of letters and numbers at the beginning of an expression or separated by either a + sign or a sign: 7xy means 7 x y. A term with no pronumeral Algebraic expression or term on each side of the equals sign. It contains an equals sign. Some important words that you need to be familiar with include: equation, expression, term, coefficient and pronumeral. When asked to find the coefficient of an algebraic term, in general it is the number in the term. If no number appears, then is assumed to be the coefficient. x = x = x In the case of a fractional term, we consider the numerical fraction as being multiplied by the pronumeral and, so, the fraction is the coefficient.

3 Chapter Introductory algebra 8 WORKED Find the coefficient of each of these algebraic terms. a 7xy b m c p d a -- 5 a Find the number in the term. The parts a The coefficient is 7. of the term are 7, x and y. b Find the number in the term. This term b The coefficient is. means one m. If no coefficient is written, a is assumed. c Find the number in the term. The parts c The coefficient is. of the term are and p. d a -- is the same as -- a. d The coefficient is Note: A term such as + does not have a pronumeral part. We call this a constant term. A constant term has no pronumeral and its value remains unchanged (constant) regardless of the value of the pronumeral. Coefficients occur only in terms where there are pronumerals and they are usually placed at the front of the term. WORKED Answer the following for each expression below. i State the number of terms. ii State the coefficient of the second term. iii State the constant term (if there is one). iv State the term with the smallest coefficient. a 4x 5xy + y b 6x xy + z + + x z a Count the number of terms. a There are 4 terms. The terms are 4x, 5xy, y and. Identify the second term ( 5xy). The The coefficient of the second term is 5. number part is the coefficient. Identify the constant term. It is the term The constant term is. with no pronumeral. 4 Identify the smallest coefficient and write the whole term to which it belongs. The term with the smallest coefficient is 5xy. b Count the number of terms. b There are 5 terms. The terms are 6x, xy, z, and x z. Identify the second term ( xy). The The coefficient of the second term is. number part is the coefficient. Identify the constant term. The constant term is +. 4 Identify the smallest coefficient and write the whole term to which it belongs. The term with the smallest coefficient is xy.

4 8 Maths Quest 9 for Victoria What is a pronumeral? A pronumeral is a letter or symbol that is used in place of a number. When we consider terms such as 7x we know from previous years that this means 7 x. The pronumeral x has a value which is not currently known or specified. Pronumerals are used to write general expressions or formulas that will allow us to make a substitution for the pronumeral when the value becomes known. When writing a general expression we choose a pronumeral that can easily be identified as belonging to the unknown quantity that it represents. WORKED Write algebraic expressions for each of the following. a A number 6 more than Ben s age b The product of a and w c One more than the age difference between Albert and his son Walter d Five times an unknown quantity is added to six times another unknown quantity. a Since Ben s age is unknown, use a a Let a = Ben s age. pronumeral. Six more means add 6. The number is a + 6. b Product means multiply. b aw c Choose pronumerals to represent Albert s age and Walter s age. c Let a = Albert s age. Let w = Walter s age. The age difference between Albert a w + and Walter is a w. Add more to this difference. d Choose pronumerals for the unknown quantities. d Let x = the first unknown quantity. Let y = the second unknown quantity. The sentence can be broken into instructions: 5 times an unknown quantity (5x)... is added to... (+) 6 times another unknown quantity (6y). 5x + 6y remember remember. A pronumeral is a letter or symbol that is used in place of a number.. A term is a combination of a number and pronumerals.. The number part of a term is called the coefficient. 4. A term that does not contain a pronumeral part is called a constant. 5. An expression is a group of terms separated by + or signs. 6. An equation is a mathematical sentence that puts two expressions equal to each other. 7. When writing expressions, think about which operations are being used, and the order in which they occur. 8. If pronumerals are not given in a question, choose an appropriate letter to use.

5 Chapter Introductory algebra 8 A Using pronumerals WORKED WORKED WORKED Find the coefficient of each of the following terms. a x b 7a c m d 8q e w f n g x y t r -- h -- i -- j In each of the following expressions state the coefficient of x. a 6x y b 5 + 7x c 5x + x d 7x x + 4 e x x f 9x x g 5x + 7x h x + 5 x x i x j 5 x k x + x + 4 l x Answer the following for each expression below. i State the number of terms. ii State the coefficient of the first term. iii State the constant term (if there is one). iv State the term with the smallest coefficient. a 5x + 7x + 8 b 9m + 8m 6 c 5x y 7x + 8xy + 5 d 9ab 8a 9b + 4 e p q 4 + 5p 7q p f 9p + 5 7q + 5p q + q g 4a + 9a b ac h 5s + s t t u i m n m + m + n j 7c d + 5d + 4 cd e 4 Write algebraic expressions for each of the following: a a number more than p b a number 7 less than q c is added to times p d 7 is subtracted from 9 times q e 4 times p is subtracted from 0 f 5 minus times p g the sum of p and q h the difference between p and q i times p is added to q j times q is subtracted from p k the product of p and q l 4 times the product of p and q m the sum of times p and times q n times p is subtracted from times q o p is divided by two times q p q is divided by p. 5 multiple choice a There are 7 students in the classroom and x are called out to see the principal. The number remaining in the room is: 7 A 7x B 7 x C x 7 D 7 + x E x b If y people enter a shop where there are customers and sales assistants, the number of people in the shop is: A y + B y C y D + y E y c If a packet of Smarties contains p Smarties, and they are to be divided up between 4 people, the number of Smarties each person receives is: p A -- B 4p C 4 + p D 4 p E p 4 4 d If a T-shirt costs n dollars, ten t-shirts would cost: n A n + 0 B 0n C D 0n + 0 E 0 n 0 Displaying the coefficients DIY SkillSHEET Mathcad.

6 84 Maths Quest 9 for Victoria Worded questions An important skill in algebra is to convert worded questions, or sentences, into algebraic expressions. Worded questions need to be read carefully so that you can decide where to place the pronumerals, coefficients and constants in an expression. Finding pronumerals, coefficients and constants The first step in converting a worded question into an algebraic expression is to identify any unknown quantities. We then identify the coefficients, constants and the arithmetic operations that connect them to form an algebraic expression. WORKED Georgia studies 4 more subjects than Henry. How many subjects does Georgia study if: a Henry studies 6 subjects b Henry studies x subjects c Henry studies y subjects. 4 a b c Read the question carefully and check for unknown quantities. All quantities are known. Henry studies 6 subjects. The number of subjects studied by Georgia is 4 more than Henry. Read the question carefully and check for unknown quantities. The number of subjects studied by Henry is the unknown x. The number of subjects studied by Georgia is 4 more than Henry. Read the question carefully and check for unknown quantities. The number of subjects studied by Henry is the unknown y. The number of subjects studied by Georgia is 4 more than Henry. a b c Number of subjects studied by Henry = 6. Number of subjects studied by Georgia = = 0 The number of subjects studied by Georgia = x + 4. The number of subjects studied by Georgia = y + 4. In other examples we may need to assign our own pronumeral to an unknown quantity. We need to explain what any pronumeral we introduce into the question represents. We should also try to use a pronumeral that is easily identifiable with what it represents.

7 Chapter Introductory algebra 85 WORKED Convert the following sentences into algebraic expressions. a If it takes 8 minutes to iron a single shirt, how long would it take to iron all of Alan s shirts? b Brenda has $5 more than Camillo. How much money does Brenda have? c In a game of Aussie rules, David kicked more goals than he kicked behinds. How many points did David score? ( goal scores 6 points; behind scores point.) 5 a b c Read the question carefully and check for unknown quantities. The number of Alan s shirts is unknown. Use a pronumeral for the unknown quantity. The total time taken is the time taken to iron shirt multiplied by the number of shirts. Read the question carefully and check for unknown quantities. The amount of money that Camillo and Brenda each have is unknown. Use pronumerals for the unknown quantities. To find the amount of money Brenda has we must add $5 to the amount that Camillo has. Read the question carefully and check for unknown quantities. The number of goals and behinds kicked by David is unknown. Use pronumerals for the unknown quantities. We need only pronumeral because there were less behinds kicked than goals. One goal is worth 6 points so multiply the number of goals by 6. One behind is worth point. Add the points from goals and behinds to find the total points scored. 4 a b c Let n = the number of Alan s shirts. The total time is 8 n = 8n. Let b = the amount of money Brenda has. Let c = the amount of money Camillo has. b = c + 5 Let g = the number of goals that David kicked. The number of behinds kicked was g. Number of points from goals = 6 g = 6g Number of points from behinds = g The number of points scored = 6g + g = 7g.

8 86 Maths Quest 9 for Victoria After converting a worded question to an algebraic expression, it is possible to see if your answer is reasonable by substituting values for the pronumerals. For example in part a of worked example 5 it takes 8 minutes to iron shirt, so it makes sense that our expression should predict that 6 minutes are needed to iron shirts. We can check by substituting n = in our expression. If n =, 8n = 8 = 6. remember remember. The first step in converting a worded question into an algebraic expression is to identify any unknowns and assign a pronumeral to each.. Worded questions need to be read carefully so that you can decide where to place the pronumerals, coefficients and constants in an expression.. We can check to see if an algebraic expression is reasonable by substituting values for the pronumerals. B Worded questions WORKED ORKED 4 Jacqueline studies 5 more subjects than Helena. How many subjects does Jacqueline study if: a Helena studies 6 subjects? b Helena studies x subjects? c Helena studies y subjects? Dianne and Angela walk home from school together. Dianne s home is km further from school than Angela s home. How far does Dianne walk if Angela s home is: a.5 km from school? b x km from school? Lisa watched television for.5 hours today. How many hours will she watch tomorrow if she watches: a.5 hours more than she watched today? b t hours more than she watched today? c y hours less than she watched today?

9 Chapter Introductory algebra 87 WORKED 5 4 Convert the following sentences into algebraic expressions. a If it takes 0 minutes to iron a single shirt, how long would it take to iron all of Anthony s shirts? b Ross has 0 dollars more than Nick. How much money does Ross have? c In a game of Aussie rules, Luciano kicked 4 more goals than he kicked behinds. How many points did Luciano score? Remember: goal scores 6 points, behind scores point. 5 Jeff and Chris play football for opposing teams, and Jeff s team won when the two teams played one another. In AFL rules, a goal scores 6 points and a behind scores point. a How many points did Jeff s team score if they kicked: i 4 goals and 0 behinds? ii x goals and y behinds? b How many points did Chris s team score if his team kicked: i 0 goals and 6 behinds? ii p goals and q behinds? c How many points did Jeff s team win by if: i Chris s team scored 0 goals and 6 behinds, and Jeff s team scored 4 goals and 0 behinds? ii Chris s team scored p goals and q behinds, and Jeff s team scored x goals and y behinds? 6 Yvonne s mother gives her x dollars for each school subject she passes. If she passes y subjects, how much money does she receive? 7 Roberto orders x cents worth of chips from a fish and chip shop, and divides them up equally between y people. What value does each person receive? 8 Brian buys a bag containing x smarties. a If he divides them equally between n people, how many does each person receive? b If he keeps half the smarties for himself and divides the remaining smarties equally between n people, how many does each person receive? 9 A piece of licorice is 0 cm long. a If David cuts d cm off, how much licorice remains? b If David cuts off -- of the remaining licorice, how much licorice has been cut off? 4 c How much licorice remains now?

10 88 Maths Quest 9 for Victoria WorkSHEET. 0 One quarter of a class of x students play tennis on the weekend. One sixth of the class play tennis and swim on the weekend. a Write an expression to represent the number of students playing tennis on the weekend. b Write an expression to represent the number of students playing tennis and swimming on the weekend. c How many students only play tennis on the weekend? During a 4-hour period, Vanessa uses her computer for c hours. Her brother Darren uses it for -- of the remaining time. 7 a For how long does Darren use the computer? b For how long do Vanessa and Darren use it altogether during a 4-hour period? Marty had a birthday party last weekend, and invited n friends. The table at right indicates Time Number of friends the number of friends at Marty s party at the 7.00 pm n 4 specified times during the evening. 7.0 pm n a How many people arrived between 7.00 pm and 7.0 pm? 8.00 pm n 8 b Between which times were the most 8.0 pm n 5 friends present at the party? c How many friends were invited but did 9.00 pm n 5 not arrive? 9.0 pm n 7 d How many friends were invited in total? 0.00 pm n e Between which times did the most friends arrive? 0.0 pm n 8 f What assumptions have been made in the previous answers?.00 pm n 4 g Write a paragraph to describe the presence of Marty s friends at his party. MATHS MATHS QUEST C H A L LL E N G G E E Is it possible to shade 6 of the 9 squares in this grid so that no three shaded squares are in a straight line (row, column or diagonal)? Jenna has read the first 7 pages of a book. When she reads 9 more pages, she will have read half the book. How many pages are in the book? If a father is now five times as old as his son, how many years ago was the son years old and the father 4?

11 Chapter Introductory algebra 89 Like terms We have already seen that a term such as xy, actually means x y. In fact, all terms consist of coefficients and pronumerals multiplied (and divided) together. In Year 7 and 8 you were introduced to like terms. Like terms contain the same pronumeral parts and can be collected (added or subtracted) if they appear in an expression. Identifying like terms Let us begin with a definition of like terms. Like terms contain the same pronumeral parts. When comparing terms, look at the pronumeral parts. If both terms contain the same combination of pronumerals (letters) they are like terms. For example 4xy and xy are like terms. Also 4xy and yx are like terms, because the order of multiplication of the pronumerals does not affect their value. The terms 5x y and xy are not like terms, even though they both contain the pronumerals x and y. This is because they contain a different combination. That is, 5x y contains x y which means x x y, whereas xy contains xy which means x y. A list of like terms for abc could include abc, 6acb, 500bca, -- abc. Can you see why? A list of terms that are not like abc could include ab, 5a bc, 6cb, ab c. Can you see why? WORKED 6 For each of the following terms, select those terms listed in brackets that are like terms. a 4y (y, y, 4x, 4xy, 4y) b 5xy ( 5xy, 5x, 5yx, 5xz, xy) c 6abc ( 6bca, 6abd, 6a bc, acb, ac b) d 7q b e 4 ( 7q b e, 6b e 4 q, 6q e 4 b, 7q 4 b e 4, 7q b e ) a The pronumeral part of 4y is y. a Like terms: y, y, 4y Check the list for terms with the same pronumeral part. b The pronumeral part of 5xy is xy. b Like terms: 5xy, 5yx, xy Check the list for terms with the same pronumeral part. c The pronumeral part of 6abc is abc. c Like terms: 6bca, acb Check the list for terms with the same pronumeral part. d The pronumeral part of 7q b e 4 is q b e 4. d Like terms: 6b e 4 q, 6q e 4 b Check the list for terms with the same pronumeral part. Collecting like terms When like terms appear in an expression they can be collected (added or subtracted). In order to decide whether to add or subtract, we look at the sign of each term. This is located on the left-hand side of each term in an expression. It is helpful to change the order of the terms before collecting them. For example, we can write the expression 4x + 5y x + 7y as 4x x + 5y + 7y. Notice that the sign on the left-hand side of each term stays the same. We can now simplify by subtracting the first terms and adding the last terms. 4x x + 5y + 7y = x + y

12 90 Maths Quest 9 for Victoria WORKED Simplify the following expressions by collecting like terms. a 6b + 5b b 6x + 5y 4x + y c 7ax + 7x 5a 6ax d 9a b ab + ab a Write the expression. a 6b + 5b Identify the like terms and simplify. = b b Write the expression. b 6x + 5y 4x + y Identify the like terms and change the order. = 6x 4x + 5y + y Simplify by collecting like terms. = x + 7y c Write the expression. c 7ax + 7x 5a 6ax Identify the like terms and change the order. = 7ax 6ax + 7x 5a Simplify by collecting like terms. = ax + 7x 5a d Write the expression. d 9a b ab + ab Identify the like terms. There are none! Cannot be simplified. 7 In the following worked example you will need to look carefully to identify the like terms. WORKED 8 Simplify the following expressions. a 6a + 9b + 7b 5b b 4a b + ba c 8ab + a b + 5a b ab a Write the expression. a 6a + 9b + 7b 5b Identify the like terms and change the order. = 6a + 9b 5b + 7b Simplify by collecting like terms. = 6a + 4b + 7b b Write the expression. b 4a b + ba Identify the like terms and change the order. = + 4a b ba Simplify by collecting like terms. = 4 6a b c Write the expression. c 8ab + a b + 5a b ab Identify the like terms and change the order. = 8ab ab + a b + 5a b Simplify by collecting like terms. = 7ab + 7a b remember remember. Like terms contain the same pronumeral parts.. When like terms appear in an expression they can be collected (added or subtracted).. It is helpful to change the order of the terms in an expression before collecting them, but be careful that the sign on the left of each term stays the same.

13 Chapter Introductory algebra 9 C Like terms WORKED 6 WORKED 7a WORKED 7b, c, d WORKED 8 For each of the following terms, select those terms listed in brackets that are like terms. a 6ab (7a, 8b, 9ab, ab, 4a b ) b x (xy, xy, 4x, 4y, yx) c az (ay, za, az, z a, a z) d x (x, x, x, x, x ) e x y (xy, xy, xy, x y, x y ) f x y 5 (xy, x 5 y, x 4 y, x y 5, x y 5 ) g 5x p w 5 ( 5x w 5 p, p x w 5, 5xp w 5, 5x p w 5, w 5 p x ) h x y 5 z 4 ( xy 5, y z 5 x 4, x + y + z, 4y 5 z 4 x, x z 4 y 5 ) Simplify the following expressions by collecting like terms. a 5x + x b y + 8y c 7m + m d q q e 7r 9r f x + 4x g a + 7a h m + m i h + 9h j 0v v k a 9a l 7p 4p m 5a + a + a n 9y + y y o 7x x + 8x p 4p p + 5p Simplify the following expressions. a m + 9m b q + 7q c 5x x d 9p p e 8m 7m f 4w w g 9x x h 5m m i r r 4 Simplify the following expressions by collecting like terms. a 6x + x y b m + n m c 9x + x x d xy + 8xy x e x + 5x x + 7 f 4m + 9m m + g 6y y 5y + 6 h 7p 6p 7p + i b b + 5b j b 6b + 8b 5 k c 5c + c + 4 l v 7v + 6v m 9h h + h + 9 n g 4g + 5g o 5m + 5m 4m + 5 p 8j + 7j 8j q 9k k + k Simplify the following expressions. a a + b + 4b b b 6m + n m + 5n c xy + y + 9yx d 5x + 7xy yx e m m f a b + 4 7ba g x + 4xy x + 7xy h x + 5x y 9x i 9a b + ba b a j ab + a b + a b ab k 9x y xy + 7yx l 4m n + n m n + 8n m x 8x + n 5xy + 9x 8yx o m n nm + 5mn p 6xy + 7x y x y 9xy 6 multiple choice What do the following expressions equal? a 8p 9p A p B p C p D p E b 5x 8x + 6x 9 A x 9 B x 9 C 5x + x 9 D 5x x 9 E 5x c a a + 5b 4b A a + b B C a b D a + b E a b d 7m n + 5m + m + m n A 9m n + 4m + B 9m n + 8 C 5m n + 8 D 5m n 4m + E 5m n + 4m + Collecting like terms SkillSHEET Mathcad.

14 9 Maths Quest 9 for Victoria Algebra rectangles The figure drawn at right is a rectangle. For the purpose of this activity x we are going to call this rectangle an algebra rectangle. The length of each algebra rectangle is x cm and the width is cm. These algebra rectangles are put together to form larger rectangles in one of two ways. Long algebra rectangle Tall algebra rectangle x x x x x x x x Let s find the perimeter of each algebra rectangle. Perimeter of -long algebra rectangle; P = (x + x + x + x + x + x) + ( + ) = 6x + Perimeter of -tall algebra rectangle; P = (x + x) + ( ) = x + 6 Find the perimeter of each of the algebra rectangles and put your results into the table below. Type of algebra rectangle Perimeter Type of algebra rectangle Perimeter -long -tall -long -tall -long 6x + -tall x long 5-long 6-long 7-long 8-long 9-long 0-long 4-tall 5-tall 6-tall 7-tall 8-tall 9-tall 0-tall Can you see a pattern? What would be the perimeter of a 0-long algebra rectangle and a 0-tall algebra rectangle? What type of algebra rectangle will have the greater perimeter, a long algebra rectangle or a tall algebra rectangle? 4 Does the answer to question depend on the value of x? 5 Suppose now that I use n tiles to make an n-tile-long algebra rectangle and an n-tile-tall algebra rectangle. Write an expression for the perimeter of each.

15 This medical first occurred in 95! Chapter Introductory algebra 9 5a + a = ab + 5b ab = ab 5ab = 0a + a 7a = ab + 8ab ab = b b b = 5a + 4b a + b = b b 4b = 4a + b a b = 6a 9a = b + b 4b b = 5a b = 7b + a 6b = ab + b ab + b = ab + ab + ab = a + a + a = 7a + b + 8b 7a = 5a + 4ab 6a = ab + 5a ab = ab + a + ab a = 7a + a 0a a = a + ab + 7a + ab = 7ab + 4a + a ab = 4a + 8ab a ab = 6ab ab ab = b + b + 7ab 4ab = ab ab ab = a + b a a = 6a + a + b 4b = 4b + ab b ab = 0ab 8ab + a = a + a + 4a = a + b a a = b + 6b 4b = 5ab + 4a 4ab = 4a 4b 5a + b = 7ab 4ab + ab = ab 7ab + ab = 7a + a a + 4a = 6a + 4b + a + b = a 5a + 4a = ab + b + ab b = 7a 8a b ab 5ab ab a 7ab b 4a a + 7ab b ab 4b 5a b b a 8a b 8b a + b 8ab a ab 0b a 6ab 6a 7b a + ab a + b a + 4ab a b 4ab b + 4ab 4a + ab 9a +6b 7a + 5ab 4a + 7b 5a + ab 5b + ab b + ab 4b + ab 8a + ab

16 94 Maths Quest 9 for Victoria For the expression 0xy + y 9 + 4z state the largest coefficient. Write an algebraic expression for a number more than x. If there are 550 people and y of them don t vote, write an expression for the number who do vote. 4 Karlie has four birds. How many will she have if she buys t more? 5 Gary has a piece of material 00 cm long. If he cuts q cm off, how long is the remaining piece? 6 multiple choice Jessie sells homemade lemonade at p cents per glass and then sells it to q people. The total amount of money, in cents, made by Jessie is: p q A p + q B pq C -- D p q E -- q p 7 One third of a class of x students watch football only and one fifth watch soccer only. Write an algebraic expression to represent the number of students who watch football or soccer. 8 Simplify the expression 4d 6d. 9 Simplify 0x + 7x 5x. 0 Simplify 9x + 4y + x y. Multiplication and division When multiplying and dividing algebraic terms, it is not necessary to have like terms. In fact any terms can be multiplied or divided and the result is a single new term. Multiplication Consider the product 8x y which equals 4xy. When we multiply the terms together we can rearrange the product and consider the multiplication of the coefficients (number parts) separately. This is because, 8x y = 8 x y = 8 x y (since order is not important) = 4 x y = 4xy (The signs still exist but are not shown.) When a pronumeral is multiplied by itself, we can use a power or index rather than writing the pronumeral each time. For example, 8x xy x = 8 x x y x = 48x y You will learn more about powers and indices later in this book.

17 Chapter Introductory algebra 95 WORKED Simplify the following. a 4a a c 5x y x b 4a b a d 7ax 6bx abx a Write the algebraic terms. a 4a a Rearrange, writing the = 4 a a coefficients first. Multiply the coefficients and pronumerals separately. = a a = a = a b Write the algebraic terms. b 4a b a Rearrange, writing the = 4 a a b coefficients first. Multiply the coefficients and pronumerals separately. = 8 a b = 8a b c Write the algebraic terms. c 5x y x Rearrange, writing the = 5 x x y coefficients first. 9 Multiply the coefficients and pronumerals separately. = 0 x y = 0x y d Write the algebraic terms. d 7ax 6bx abx Rearrange, writing the coefficients first. = 7 6 a a x x x b b Multiply the coefficients and = 84 a x b pronumerals separately. The = 84a b x simplified term is often written with the pronumerals in alphabetical order. Division When dividing terms, we write the division as a fraction and try to simplify by cancelling the numerator and denominator by any common factors. The coefficients (number parts) and pronumeral parts can be treated separately. 6xy For example, 6xy x would be written as: x 6 x y = x y = = y

18 96 Maths Quest 9 for Victoria Simplify the following. 6x 4xy a b c 8ab 6a b d 5xyz 9x 0 yz 6x a Write the term. a x Cancel 6 and (common factor of ). = = x 4xy b Write the term. b yz c d WORKED 0 Cancel 4 and 0 (common factor ). Cancel y from numerator and denominator. Write the terms and express as a fraction. The term a means aa. Cancel 8 and 6 (common factor 8). Cancel a and b from numerator and denominator. Write the terms and express as a fraction. The term x means xx. We cannot cancel 5 and 9. Cancel x from numerator and denominator. = x z c 8ab 6a b 8ab = a b 8ab = aab = a d 5xyz 9x 5xyz = x 5xyz = xx 5yz = x remember remember. When multiplying and dividing algebraic terms it is not necessary to have like terms.. For multiplication we can multiply the coefficients (number parts) and the pronumeral parts separately.. A division problem should be expressed as a fraction. 4. For division, try to simplify by cancelling the numerator and denominator by any common factors.

19 Chapter Introductory algebra 97 D Multiplication and division WORKED 9 WORKED Simplify the following. a m n b 4x 5y c p 4q d 5x y e y 4x f m 5n g 5a a h 4y 5y i 5p p j m 7m k mn p l 6ab b m 5m mn n 6a ab o xy 5xy x p 4pq p q q 4c 7cd c r a 5ab ab Simplify the following. 0 6x 9m y a b c d 8m e m f 4x 7 g x h m 8 i 4m j 6x 8mn 6xy k l n y m 6ab 8xyz n o m a b 4x p 70ab x q yz b 8xz r 7xy z xyz Simplify the following. a 5x 4y xy b 7xy 4ax y c x 4xy yx 6x d y 5x e ab p f q y b x p q g 4a 5ab a h a 4ab ba b i a a a a SkillSHEET Multiplication and division WorkSHEET. Mathcad. MATHS MATHS QUEST C H A L LL E N G G E E Place two different sets of coloured counters (or coins) on the grid as shown. Show how you can swap the position of the two sets of counters in exactly 8 moves. A counter can slide into an empty square next to it or can jump over another counter into an empty space. Record your solution. Try this game again but with two different sets of counters and a grid of 7 squares. Try to swap the two sets of counters in exactly 5 moves. Record your solution.

20 98 Maths Quest 9 for Victoria = a x 5a = = 7b x b x b = = 5a x 7e = = b x a = = b x b x = = c x a x c = = c x c x c = = 5e x e = = 7e x a = = c x a = = e x c x e = = 6a x b x a = = e x e x e = = 5c x 5c = 9c 6c 4 ac 5ce 5b 4be = 0ae 5e = = 0ae 6a = = 0a e 4e = = a c 6a = 6bc = = 8b 0abc = = ac 5a = e = 5a 50a = b = 0ab = a b b = 4a = b = 8ab = 00c e 0c = = b e be = = c x c = 0a = c = a = a x 8a a = = b x e x 7 = 8c = a = ca = e x e x e = = a b b = = a x 4b = 64b = e = 6b e = x a x 4e = 5c = e = 5c 5a 6ab 4a ae ac 5ae ae 5ab 6ce 0e 8e 5e 4b 6e 8ab 5ac c a 4be 5a 4ae 4ac a 5c 6a 7ce 7b b 6c a a b

21 Chapter Introductory algebra 99 Algebraic fractions Algebraic fractions contain pronumerals that may represent particular numbers or changing values. The methods for dealing with algebraic fractions follow the same principles that we 5 4 used for numerical fraction questions, such as or That is, find a common denominator for addition and subtraction, or use reciprocals for division. Adding and subtracting algebraic fractions To add or subtract algebraic fractions we perform the following steps. Step Find the lowest common denominator (LCD) by finding the lowest common multiple (LCM) of the denominators. Step Rewrite each fraction as an equivalent fraction with this common denominator. Step Add (or subtract) the new numerators. WORKED Simplify each of the following expressions. x x y y y + x+ y a b c x x a Write the expression. a x = Find the lowest common denominator (LCD). The lowest common multiple (LCM) of and 5 is 0. Add the numerators. 7x = b Write the expression. b y y Find the LCD. The LCM of and 7 is. 7y 6y = Subtract the numerators. y = c Write the expression. c y + x + y Find the LCD. The LCM of 5 and 6 is 0. 6( y + ) 5( x + y) = Add the numerators. 6( y + ) + 5( x + y) = Expand the brackets in the numerator. 6y x + 5y = Simplify the numerator by collecting like terms. 5x + y + 6 = x

22 00 Maths Quest 9 for Victoria If pronumerals appear in the denominator we can treat these separately to their coefficients (numbers). In such a case the lowest common denominator (LCD) is found by finding the lowest common multiple (LCM) of the coefficients, then including in the LCD every pronumeral used. WORKED Simplify the following expressions. x 6 a b c x x y 5 y 5z x a Write the expression. a -- x Find the LCD. The LCM of and is. The only pronumeral is x, so include it in the LCD. The LCD is x. Subtract the numerators. = x = x x b Write the expression. b y Find the LCD. The LCM of and 5 is 5. The only pronumeral is y so include it in the LCD. The LCD is 5y. 0x = y 0x + 9 Add the numerators. = y 6 c Write the expression. c z Find the LCD. The LCM of 5 and 0 is 0. The only pronumeral is z, so include it in the LCD. The LCD is 0z x y x x = z y xz Subtract the numerators. = z xz z Multiplying and dividing algebraic fractions The rules for multiplication and division are the same as for numerical fractions. When multiplying algebraic fractions, multiply the numerators and multiply the denominators, then cancel any common factors in the numerator and denominator.

23 WORKED Check for common factors in the numerator and denominator and cancel. The numbers and 6 have a common factor of. Cancel y. Chapter Introductory algebra 0 Simplify each of the following. a x 6 y z b y x y a Write the algebraic fractions. a x y Multiply the numerators and multiply the denominators. 6x = y Check for common factors in the numerator and denominator x = and cancel. The numbers 6 and have a common factor of. y b Write the algebraic fractions. b y z x y Multiply the numerators and multiply the denominators. yz = xy = z x When dividing algebraic fractions, change the division sign to a multiplication sign and write the following fraction as its reciprocal. This is the same times and tip method that was covered in chapter using numerical fractions. WORKED 4 Simplify each of the following. 4 xy 4x a b x x 9 y 4 a Write the algebraic fractions. a x x Change the division sign to a multiplication sign and write the = -- second fraction as its reciprocal. x x Multiply the numerators and multiply the denominators. = x Check for common factors in the numerator and denominator. 4 = -- Cancel x. 4 xy 4x b Write the algebraic fractions. b y Change the division sign to a multiplication sign and write the xy = second fraction as its reciprocal. 7xy Multiply the numerators and multiply the denominators. = x 4 Check for common factors in the numerator and denominator. 7y = Cancel x. 8 x y x

24 0 Maths Quest 9 for Victoria remember remember. Algebraic fractions contain pronumerals that may represent particular numbers or changing values.. To add or subtract algebraic fractions, we use the same method as with numerical fractions. (a) Find the lowest common denominator (LCD) by finding the lowest common multiple (LCM) of the denominators. (b) Rewrite each fraction as an equivalent fraction with this common denominator. (c) Add (or subtract) the new numerators.. When multiplying algebraic fractions, multiply the numerators, and multiply the denominators. Cancel any common factors in the numerator and denominator. 4. When dividing algebraic fractions, change the division sign to a multiplication sign and write the following fraction as its reciprocal (swap the numerator with the denominator). Continue as for multiplying algebraic fractions. E Algebraic fractions SkillSHEET SkillSHEET Mathcad SkillSHEET.4 WORKED.5 Algebraic fractions WORKED WORKED.6 Simplify each of the following expressions. x x y y m a b c x x m m t d e f a a p 5 p 4q g h i x x j Simplify each of the following expressions. a b c p p x 5x 4m d e f b 4b 6c 9c y Simplify each of the following a b c x 9 4 y d e f x y 4 m n 7m 0 g h i m 5 x 0 6 x 5 j k l x 5 y 5 6 4m 9 7 p x m n o m 5 p m t -- 5 q m y 6 -- x

25 Chapter Introductory algebra 0 WORKED 4 4 Simplify each of the following. 5 5 a b c x x 4 d e f m m a a 6 0 a a g h i b b 4 7 6m ab a j k l b m 0m 0 x y m n o p 9 pq 5 m m 8 5 y -- 4 SkillSHEET Algebra 00.7 GAMEtime Write down an algebraic expression for the sum of 5 times m and 4 times n. Ben has 6 kittens. How many will he have if he sells x of them? Karen has a bread stick 60 cm long. If she cuts off p cm, how much bread remains? 4 Simplify 4y 7y + 5y +. 5 Simplify m 4n. 8x 6 Simplify x x 7 Simplify xy 0z 8 Simplify x 9 Simplify b 5b x 4x 0 Simplify y yz Substitution and formulas In mathematics, science and engineering, algebraic expressions and formulas are commonly used. For example, in chapter you learned the formula for Pythagoras theorem (c = a + b ) which enabled you to find the unknown side in a right-angled triangle. In this section we will look at how to substitute particular values for the pronumerals in an expression or formula.

26 04 Maths Quest 9 for Victoria Substitution We can evaluate (find the value of) an algebraic expression if we replace the pronumerals with their known values. This process is called substitution. Consider the expression 4x + y. If we substitute the known values x = and y = 5, we obtain = = Rather than showing the multiplication signs, it is common in mathematics to write the substituted values in brackets. We would write the example above as: 4x + y = 4() + (5) = = WORKED 5 If x = and y =, evaluate the following expressions. a x + y b 5xy x + c x + y a Write the expression. a x + y Substitute x = and y =. = () + ( ) Evaluate. = 9 4 = 5 b Write the expression. b 5xy x + Substitute x = and y =. = 5()( ) () + Evaluate. = = 8 c Write the expression. c x + y Substitute x = and y =. = () + ( ) Evaluate. = = Substitution into formulas A formula expresses one quantity in terms of one or more other quantities. For example, the formula for the area of a rectangle is given by: Area = length width or A = l w. If a particular type of kitchen tile has a length, l = 0 cm, and width, w = 5 cm, we can substitute these values into the formula to find its area. A = l w = 0 5 = 00 cm The same formula can be used to calculate the area of a tile of different size (provided it is rectangular in shape) by substituting whatever the length (l) and width (w) happen to be. 5 cm 0 cm

27 WORKED 6 Chapter Introductory algebra 05 Ivan, the electrician, knows that the formula for the voltage in an electrical circuit can be found using the formula known as Ohm s Law: V = IR where I = current in amperes, R = resistance in ohms and V = voltage in volts. Find V when: a I = amperes, R = 0 ohms b I = 0 amperes, R = 0 ohms c I = 0.6 amperes, R = 6600 ohms. a Write the formula. a V = IR Substitute I = and R = 0. = ()(0) Evaluate and express the answer = 0 volts in the correct units. b Write the formula. b V = IR Substitute I = 0 and R = 0. = (0)(0) Evaluate and express the answer = 00 volts in the correct units. c Write the formula. c V = IR Substitute I = 0.6 and R = = (0.6)(6600) Evaluate and express the answer in the correct units. = 960 volts The distance (x) travelled by an object is given by the formula: x = ut + -- at, where u = the starting speed in m/s, a = acceleration in m/s and t = time in seconds. A car starts at a speed of 50 m/s and accelerates at 6 m/s for 7 s. How far has the car travelled? WORKED 7 Write down the formula. x = ut + at Substitute u = 50, a = 6, and t = 7. = (50)(7) + -- (6)(7) Evaluate and express the answer in the correct units. = = 497 m remember remember. We can evaluate (find the value of) an algebraic expression if we substitute the pronumerals with their known values.. Rather than showing the multiplication signs, it is common in mathematics to write the substituted values in brackets.. Formulas can be used to calculate quantities when known values are substituted. --

28 06 Maths Quest 9 for Victoria F Substitution and formulas EXCEL EXCEL SkillSHEET Spreadsheet Mathcad Spreadsheet.8 WORKED 5 Substitution Substitution Substitution game WORKED 6 WORKED 7 If x = 4 and y =, evaluate the following expressions. a 4x + y b xy x + 4 c x y The formula for the voltage in an electrical circuit can be found using the formula known as Ohm s Law: V = I R where I = current in amperes, R = resistance in ohms and V = voltage in volts. Find V when: a I = 4 amperes, R = 8 ohms b I = 5 amperes, R = 0 ohms c I = 0.8 amperes, R = 00 ohms d I =.5 amperes, R = 70 ohms Evaluate each of the following by substituting the given values into each formula. a If A = bh, find A when b = 5 and h =. b m If d = --- find d when m = 0 and v =. v c If A = -- xy, find A when x = 8 and y =. d If A = -- (a + b)h, find A when h = 0, a = 7 and b =. AH e If V = , find V when A = 9 and H = 0. f If v = u + at, find v when u = 4, a =. and t =.. g If t = a + (n )d, find t when a =, n = 0 and d =. h If A = -- (x + y)h, find A when x = 5, y = 9 and h =.. i If A = b, find A when b = 5. j If y = 5x 9, find y when x = 6. k If y = x x + 4, find y when x =. l If a = b + 5b, find a when b = 4. m If s = ut + -- at find s when u = 0.8, t = 5 and a =.. mp n If F = find F when m = 6.9, p = 8 and r =. (answer to r decimal places). o If C = π d, find C when π =.4 and d =. 4 a The area of a triangle is given by the formula A = -- bh, where b is the length of the base and h is the height of the triangle. Find A when: i b = 6 cm and h = 4 cm ii b = 5 cm and h = cm iii b = cm and h = cm. b The formula to convert degrees Fahrenheit (F) to degrees Celsius (C) is 5 C = -- (F ). Find C when: c 9 i F = 59 ii F = 44 iii F =. The length of the hypotenuse of a right-angled triangle (c) can be found using the formula, c = a + b b are the lengths of the other two sides. Find c when: i a = and b = 4 ii a = and b = 5 iii a = 8 and b = 6., where a and

29 Chapter Introductory algebra 07 d If the volume of a prism (V) is given by the formula V = Ah, where A is the area of the cross-section and h is the height of the prism, find V when: i A = 7 cm and h = 9 cm ii A = 0 cm and h = 58 cm iii A =.6 cm and h =. cm. e Find the number of edges (E) on a prism if it can be found using E = F + V where F is the number of faces and V is the number of vertices when: i F = 5 and V = 7 ii F = 7 and V = 0 iii F = 0 and V =. f The kinetic energy (E) of an object is found by using the formula E = -- mv where m is the mass and v is the velocity of the object. Find E when: i m = and v =.6 ii m = 5 and v =.6 iii m = 0. and v = 0. g The volume of a cylinder (v) is given by v = π r h where r is the radius and h is the height of the cylinder. Find v if π =.4 and: i r = 7 and h = ii r = 49 and h = 9. (to decimal places) iii r =.5 and h =.98 (to decimal places). h The surface area of a cylinder (S) is given by S = πr (r + h). Find S (to decimal places) if r is the radius of the circular end, h is the height of the cylinder and π =.4, when: i r = 4 and h = 5 ii r = and h = 0 iii r =.4 and h = 7.. Good health At the beginning of this chapter we considered the case of Richard who is 86 kg and.75 m tall. We know that the Body Mass Index (B) can be found using the m formula B = ----, h where m is the person s mass, in kilograms, and h is the height, in metres. Calculate Richard s Body Mass Index. A person is considered to be in a healthy weight range if B 5. Comment on Richard s weight for a person of his height. Calculate the Body Mass Index for each of the following people: a Judy who is.65 metres tall and has a mass of 5 kilograms b Karen who is.78 metres tall and has a mass of 79 kilograms c Manuel who is.7 metres tall and has a mass of 85 kilograms. 4 Calculate your own Body Mass Index. Algebra 00 WorkSHEET GAMEtime.

30 08 Maths Quest 9 for Victoria History of mathematics N km HERON (c. 00 AD) and BRAHMAGUPTA ( AD) Greece Alexandria Heron lived here. Turkey Baghdad Heron and Brahmagupta were mathematicians who developed formulas for finding the area of geometrical shapes. Heron Heron, also called Hero, was a Greek mathematician and inventor. He lived in Alexandria in Egypt in the first century AD. Translations of his works still survive. His interests included numbers, geometry, astronomy and mechanics. He invented many machines and toys such as fountains, syphons and steam engines which were operated by water, steam or compressed air. In geometry his work included methods of finding the areas of regular shapes and polygons. He is credited with the discovery of a formula for finding the area of a triangle. Heron s formula states that for any triangle with sides a, b and c then the area of the triangle is ss ( a) ( s b) ( s c) a b a+ b+ c where s = c Heron s formula can be used to prove Pythagoras theorem. Brahmagupta Brahmagupta was a mathematician and astronomer who lived during the seventh century AD in north west India. He wrote several books but his most important work was called Brahma-sphuta-siddhanta which means the opening of the universe. It described the Hindu astronomical system. Two of its 5 chapters are devoted to mathematics. Brahmagupta They include work on lived here. arithmetic progressions, quadrilaterals, right-angled triangles, volumes and surfaces. Brahmagupta developed a formula for the maximum area of a quadrilateral where for any quadrilateral of side lengths a, b, c and d the a maximum area is ss ( a) ( s b) ( s c) ( s d) d b a+ b+ c+ d where s = c It is interesting to note that this is the same as Heron s formula with one of the four side lengths set to zero. A general case for a quadrilateral A general formula for the area of any quadrilateral is ( s a) ( s b) ( s c) ( s d) abcd cos A+ C a+ b+ c+ d where s = and A and C are the internal angles, or -- 4 p q (b + d a c ) 4 where a, b, c and d are the side lengths and p and q are the diagonal lengths. Questions. Name one of Heron s inventions.. What does Heron s formula determine?. What was Brahmagupta s job? 4. What does Brahmagupta s formula determine? 5. What is the connection between the formulas of Heron and Brahmagupta?

31 Chapter Introductory algebra 09 summary Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows. A is a group of letters and numbers within an expression. The number in the term 6xy is called a. The letter in the term 7a is called a. 4 An is an algebraic expression that contains an equals sign. 5 7abc and 6bca are called terms. 6 The expression 7x + 5y + x 6y is equivalent to. 7 The first step in converting a into an algebraic expression is to identify any pronumerals. 8 When appear in an expression they can be collected (added or subtracted). 9 When adding or subtracting algebraic fractions, you first must find the. 0 When multiplying algebraic fractions, multiply the numerators together and the together. Try to simplify by the numerator and denominator by any common factors. When dividing algebraic fractions, change the division sign to a sign and write the following fraction as its. We can evaluate an algebraic expression if we the pronumerals with their known values. WORD LIST cancelling worded question coefficient term equation reciprocal 0x y like terms like pronumeral substitute LCD multiplication denominators

32 0 Maths Quest 9 for Victoria CHAPTER review A B C D E F F test yourself CHAPTER a For the expression 8xy + x + 8y 5: i state the number of terms ii state the coefficient of the first term iii state the constant term iv state the term with the smallest coefficient. b Write expressions for the following, where x and y represent numbers: i a number 8 more than y ii the difference between x and y iii the sum of x and y iv 7 times the product of x and y v times x is subtracted from 5 times y. a Leo receives x dollars for each car he washes. If he washes y cars, how much money does he earn? b A piece of rope is 4 metres long. i If George cuts k m off, how much is left? ii After George has cut k m off, he divides the rest into three pieces the same length. How long is each piece? Simplify the following expressions by collecting like terms. a 8p + 9p b 7m + m + m c 5y + y 4y d 8ab + b + ba e 9s t s t f 5x + 6xy x + xy g c d cd + 5dc h 7x y 8 x y + i n p q p q + 6 j 8ab + a b 5a b + 7ab 4 Simplify the following expressions. a 6a b b 5y y c ab b d p pq e xy 4yx 0a 4x f g h i 8 4b j a 5a 4m 5 Simplify the following expressions. x x x x x 5x 5 7 a a b c d e f m 8m 6 a y 7 x 6 p a 4 0 a a g h i j x 4 p 5 5 a m 5m 0a 4ab k l m n m m 7y y 6y xy 4c 7c 6 If y = 5x + x, find y when: a x = b x = 5. 7 The volume (v) of each of these paint tins is given by the formula, v = πr h, where r is the radius and h is the height of the cylinder. Find v (to decimal places) if π =.4. a r = 7 cm and h = cm b r = 9. cm and h = 9.8 cm. y -- 8