2To raise money for a charity,

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1 Algebra an equations To raise money for a charity, a Year 0 class has ecie to organise a school ance. Tickets to the school ance will cost $6 each. Expenses have been calculate as $00 for the hire of the venue an $0 for a DJ. The ance organisers hope to raise more than $000 for charity. How many people nee to atten the ance to achieve this goal? One way to come up with an answer to this problem is to solve an inequation. In this chapter we will look at equations an inequations an consoliate an exten a number of your algebraic skills.

2 0 Maths Quest 0 for Victoria Operations with pronumerals Like terms are terms that contain the same pronumerals an can be collecte (ae or subtracte) in orer to simplify an algebraic expression. Simplify the following. a j c + c + j b + 9 a Write the expression. a j c + c + j Ientify the like terms an group j + j c + c them together. Simplify by collecting like terms. 7j c b Write the expression. b + 9 Simplify by collecting like terms. When multiplying an iviing algebraic terms, it is not necessary to have like terms. In fact, any terms can be multiplie or ivie an the result is a single new term. Simplify the following. 8mc a m 6p b c 0mf a b - 6ab a Write the expression. a m 6p Rearrange, writing the coefficients 6 m p first. Multiply the coefficients an pronumerals separately. 0mp b Write the expression. b Cancel 8 an 0 (common factor of ) an cancel m from numerator an enominator. c Write the expression. c Cancel a from numerator an enominator. Cancel b from numerator an enominator. 8mc 0mf 9c f a b - 6ab a - 6b

3 remember remember Chapter Algebra an equations. Like terms contain the same pronumeral parts an can be collecte (ae or subtracte) in orer to simplify an algebraic expression.. When multiplying an iviing algebraic terms, it is not necessary to have like terms. In fact, any terms can be multiplie or ivie an the result is a single new term. A Operations with pronumerals Simplify the following. a k + k + c + 9c b 6m + 9m + 0f + 6f c + c + + c f + h + f + h e 7g + j + g + j f g 9n + + n + 7 h y + 6h + h + 9y i nv + 8u + 7nv + u SkillSHEET. Simplify the following. a 0m 7m + c c b a 6a + f 8f c k + k 9 7t t + 7 e 0r + r 0 f 6v 8v + 7 g 6p 9 p h 0w 6w + 0 i c 8 + c 9 j j + c j c k k + m k 0m l + c c m y + y + y 7y n x x + x 9x o c c SkillSHEET. a multiple choice a q + 6p q p simplifies to: A 7q + p B 7q p C q + p D q p E p 7q b r 6y y + r simplifies to: A 9r 9y B r 9y C r + 9y D 9r + 9y E r + 9y c j 6j j + j simplifies to: A j j B j j C j j D j 6 E j j Simplify the following. a x + x + x + 6 b c v v 8v 6 a ab + ab + b e u + u u f n n + 6n Simplify the following. a 8f h b 0ab 6c c m 7g 6p 9hn e b 6at 7s f ma 6t hs 6 Simplify the following. b, c jkl mnt a - b - 8 jl 6t c 6ab 7kg - e bc 6kh f g x yz mn - h k - xyz m n i kgh 7kh 0a b ab abc - b Operations with pronumerals SkillSHEET Mathca.

4 Maths Quest 0 for Victoria Substituting into expressions When the numerical values of pronumerals are known, we can substitute them into an algebraic expression an evaluate it. It can be useful to place any substitute values in brackets when evaluating an expression. If a, b, an c 7, evaluate the following expressions. a a b b a + 9b c a Write the expression. a a b Substitute a an b into the expression. Simplify. b Write the expression. b a + 9b c Substitute a, b an c 7 () + 9() ( 7) into the expression. Simplify If c a + b, fin c if a an b. Write the expression. c a + b Substitute a an b into the ( ) + ( ) expression. Simplify. + remember. When the numerical values of pronumerals are known, we can substitute them into an algebraic expression an evaluate it.. It is sometimes useful to place any substitute values in brackets when evaluating an expression. 69

5 Chapter Algebra an equations B Substituting into expressions If a, b an c, evaluate the following expressions. a a + b b c b c c a b c (a b) e 7a + 8b c f a b c + + g abc h ab(c b) i a + b c j c + a k a b c l.a.b If 6 an k, evaluate the following. a + k b k c k k e (k + ) f g k h k i k If x an y, evaluate the following. a x + y b y x c xy x e x y 9x f - y y Complete the following. a If c a + b, calculate c if a 8 an b. b If A bh, fin the value of A if b an h. c The perimeter, P, of a rectangle is given by P L + B. Fin the perimeter, P, of a rectangle given L.6 an B.. C If T -, fin the value of T if C 0. an L.. L e n + If K -, fin the value of K if n. n f Given F 9C +, calculate F if C 0. g If v u + at, evaluate v if u 6, a, t 6. h The area, A, of a circle is given by the formula A πr. Calculate the area of a circle, correct to ecimal place, if r 6. i If E mv, calculate E if m, v. A j Given r -, evaluate r to ecimal place if A 68. π SkillSHEET Substituting into expressions EXCEL Substitution EXCEL Substitution game. Mathca Spreasheet Spreasheet multiple choice a If p an q, then pq is equal to: A 0 B C D 0 E b If c a + b, an a 6 an b 8, then c is equal to: A 8 B 00 C 0 D E c Given h 6 an k 7, then kh is equal to: A 9 B C 76 D 776 E 8

6 Maths Quest 0 for Victoria Career profile KARIN XUEREB Supervising Meteorologist Qualifications: Bachelor of Applie Science Diploma of Meteorology Master of Applie Science Employer: Bureau of Meteorology Company website: Ever since I was very young, I have striven to unerstan nature an the universe. Working in a scientific fiel allows me to have pai work that is fun an enjoyable. I am able to eal with scientific problems that are like a puzzle that is challenging, but very rewaring when the pieces fit together. The esire to unerstan the secrets of nature is what rew me to this fiel. It is ifficult to outline a typical ay s work because the work can vary epening on the type of project I am working on (which changes from time to time). Generally, I examine meteorological charts an meteorological ata to unerstan the meteorology of past significant rainfall events, write reports summarising finings, write computer programs to analyse meteorological ata, write scientific papers for publication in scientific journals or for presentation at conferences, an rea scientific journals. I frequently use mathematics in my work, for example, in areas such as linear interpolation (fining the value of a point on a graph between two known points) an weighte averages of temperatures. I also work with many algebraic equations or formulas. For example, to work out the temperature of a ry air parcel, if brought to the surface of the Earth, the following formula woul be use: P s P P T s T P - R c p where T s is the temperature at the Earth s surface T P is the temperature at a given air pressure P s is the air pressure at the Earth s surface (approximately 000 hectopascals) P P is the given air pressure R is the gas constant for ry air c p is the specific heat at constant pressure. R For ry air, c p So, if given the temperature at a pressure of 900 hectopascals (approximately km above the Earth s surface), the temperature of a ry air parcel, if brought to the surface woul be: T s T Maths is the key to unlocking the secrets of nature. It is fun an sometimes challenging, an I enjoy challenges. Questions. What is the approximate air pressure at the Earth s surface?. How many pascals are there in a hectopascal?. Fin out how to become a meteorologist.

7 Chapter Algebra an equations Expaning Expaning brackets in an algebraic expression is achieve by multiplying the term outsie the brackets by each of the terms insie. For example, (x + y) x + y x + 8y Remember that rather than writing multiplication signs ( ), we can write the pronumerals in brackets. In this case we coul have written: (x + y) (x) + (y) x + 8y When more than one set of brackets appears in an expression, we can often simplify by collecting any like terms that result from expaning the brackets. For example: (x + y) + (x y) (x) + (y) + (x) + ( y) x + 0y + 6x y 8x + 7y Expan: a 7(m ) b 6(a ). a Write the expression. a 7(m ) Multiply each term insie the 7(m) + 7( ) brackets by the term outsie. 7m 8 b Write the expression. b 6(a ) Multiply each term insie the 6(a) 6( ) brackets by the term outsie. 6a + 8 Expan an simplify 6(m r) (m + 7r). 6 Write the expression. 6(m r) (m + 7r) Multiply each term insie the brackets by the term outsie. 6(m) + 6( r) (m) (7r) 6m r m r Simplify by collecting like terms. m 8r remember remember. Expaning brackets in an algebraic expression is achieve by multiplying each term insie the brackets by the term outsie.. When more than one set of brackets appears in an expression we can often simplify by collecting any like terms that result from expaning the brackets.

8 6 Maths Quest 0 for Victoria EXCEL GC Mathca Spreasheet program SkillSHEET Expaning a Expaning b Expaning 6. C Expaning Expan the following. a (k + ) b 7(m + ) c (y + 7) 8( 9) e (h ) f (k 6) g (m ) h (6t + ) i 8(k ) j (m + n) k 8(y f ) l 6(v + 7w) m b(c ) n k(i + ef ) o 6p(j m) Expan the following. a (c + ) b ( + ) c 6(m + ) 8(c + ) e (k m) f 7( x) g 0( y) h k(k + ) i x(x ) Expan an simplify. a (c + ) + (c + ) b (k + ) + (k + 6) c 8(m + ) + (m + ) ( j ) ( j + ) e 7(t + ) (t + ) f 9(m + 7) + (m 6) g 0(c + ) + 6(c 9) h 6( ) ( ) i (w ) 8(w + 8) j (h + 7) 0(h ) k (y 8) + (y ) l (x + ) (x ) m 0(h ) (h ) n c + + (c + 7) o (m 9) (m ) Expan an simplify. a y(y 6) + (y 6) b w(w + ) 6(w + ) c x(x ) (x ) h(h + ) + (h + ) e f (f + ) 8(f + ) f a(a ) + (a ) GAMEtime WorkSHEET Algebra an equations 00. multiple choice a y( y) simplifies to: A y B 6 y C y 6y D y y E y y b (k ) simplifies to: A k B k + C k + 8 D k 8 E k + c (b ) (b ) simplifies to: A b 6 B b + 8 C b D b E b 8 Simplify r t + r t. 6e Simplify. 0e Simplify gh gkm. If l 7 an m, evaluate l 6m -. m If c 0.0 an 0., evaluate 6c. 6 If c a + b, evaluate c when a an b 6. Leave your answer in simplifie sur form. 7 Expan 6(w v). 8 Expan q(p q). 9 Expan an simplify ( + u) + (8u + 7). 0 Expan an simplify 7( r) (r + ).

9 Chapter Algebra an equations 7 Factorising using common factors Factorising is the opposite process to expaning. In this section, we will look at factorising by taking out the highest common factor (HCF) of an algebraic expression. (Further factorising techniques will be covere in chapter, Quaratic equations.) Factorise the following. a 6a b 0p 6 + p 7 a Write the expression. a 6a Fin the highest common factor (HCF) of the terms. Place the HCF in front of a set of (a ) brackets an ivie each term by the HCF to complete the insie of the brackets. b Write the expression. b 0p 6 + p Fin the highest common factor (HCF) of the terms. Place the HCF in front of a set of p (p + ) brackets an ivie each term by the HCF to complete the insie of the brackets. HCF HCF p remember remember. To factorise an expression: (a) Fin the highest common factor (HCF) of the terms an place in front of the brackets. (b) Divie each term by the HCF an place insie the brackets.. If the first term is negative, take out a negative common factor. D Factorising using common factors SkillSHEET.6 7a Factorise the following. a x + b 6y + c 7m + 9 0y + 0 e f + 8h f a 9 g b h 6 6 i 8e j 6l 7 k n 6p l 7f 98 Factorising Mathca

10 8 Maths Quest 0 for Victoria 7b Factorise the following. a 6t + 0 b 9m + 6 c k + 8 0m + e m + n f 0j g 6c 7 h 00h i 0m j c + c k 6ak 0am l abc + bc Factorise the following (by taking out a negative common factor). a c + b 7m + c 8k + j j 0 e h 8j f 6p s g 9k + h 6ac + a i bm 0abc Factorise the following. a m + m b 6 c x + 6x 8f + f e y y f 7p + p g q 0q 8 h r r i a b + ab j 0m n mn k 6k p + 8k p l x y xy multiple choice a b c 0h + factorises to: A (h + 6) B 0(h + ) C (h + 0) D (h + 6) E (h + 6) 8 + r factorises to: A ( r) B ( r) C ( + r) D ( + r) E ( r) When fully factorise, 8au 0uh factorises to: A (au uh) B 8u(a.h) C u(a + h) D 8(au.h) E u(a h) 0a n + a 7 n factorises to: A 6(a n + a 7 n ) B 6an(a n + a 6 n ) C 6a n (an + a n) D 6a n (n + a ) E a n ( + a ) MATHS MATHS QUEST C H A L LL E N G G E E The lines in the iagram at right show A paths between A an B. The istance from A to B along an outsie ege is 8 units. Starting at A an ening at B, what is the longest istance you coul travel along the given paths without travelling along any line more than once? Which numbers less than 00 have exactly factors (incluing one an B the number itself)? Which numbers less than 00 have exactly factors? Which number less than 00 has exactly 7 factors? Which number or numbers less than 00 have the largest number of factors?

11 This is preicte to occur in 06. Fin common factors to factorise the expressions given to fin the answer coe. Chapter Algebra an equations 9 F 8 x H 6 8x A x 0 E 6x 8 M x 8x T x x x C ae + a L x R cx c E x 0x E x + x N 6x x U e 6e L 9e + 8 F c Y 6e + e A x 9 S x + H 0x 0x I aec ac T 6x + x E ex + x V 8x x + x S ae + ac T ce + e E 0ex + e O x + x R e + e Y x + A x x T x x E 6x x + N c c V x S 8a a F x 6x + O x + 0x Y x + E x 7x T x + R 6ce c E x E 7ae R x 6x x (x + ) 0x (x ) e (x + ) e (e + ) x (x + ) x (x x ) e (e ) c (x ) x ( x) x (x + ) (c ) 8 (x + ) (x ) 9 (e + ) (x + ) x (x 7) (x + ) (x ) a (e + ) x (x ) x (x ) (x + ) x (x ) (x + ) 6 ( x) (x + ) (x + ) c (e ) a (8 a) x (x ) 7 (x ) 7 (ae ) c (6c ) e (c + ) (x + ) (x + x ) ac (e ) x (x x + ) x ( e ) e (e ) (x x + ) x (x ) x (x ) a (e + c)

12 0 Maths Quest 0 for Victoria Aing an subtracting algebraic fractions The methos for ealing with algebraic fractions are the same as those use for numerical fractions. To a or subtract algebraic fractions we perform the following steps.. Fin the lowest common enominator (LCD) by fining the lowest common multiple (LCM) of the enominators.. Rewrite each fraction as an equivalent fraction with this common enominator.. Express as a single fraction.. Simplify the numerator. Simplify the following expressions. x x x + x + a - b a Write the expression. a Rewrite each fraction as an equivalent fraction using the LCD. The LCM of an is 6. x ( ) ( x) Express as a single fraction. x x - 6 Simplify the numerator. x 6 b Write the expression. b 8 Rewrite each fraction as an equivalent fraction using the LCD. The LCM of an is. ( x + ) ( x + ) ( x + ) + ( x + ) Express as a single fraction. Simplify the numerator by expaning brackets an collecting like terms. x x - x + x x + + x + 0-8x + -

13 Chapter Algebra an equations Simplify - -. x x If pronumerals appear in the enominator, we can treat these separately to their coefficients. You can see how this is one in the next worke example. Write the expression. 9 Write equivalent fractions using the LCD. The LCM of an is. The LCM of x an x is x. So the LCD is x. - - x x 8 x x Express as a single fraction. 8 - x Simplify the numerator. x When there are two or more terms in the enominator of each fraction, we can obtain a common enominator by writing the prouct of the enominators. For example, the lowest common enominator of x + an x is simply (x + )(x ). x + x Simplify - by writing it first as a single fraction. x + + x + Write the expression. 0 Write equivalent fractions using the LCD. The LCM of x + an x + is the prouct (x + )(x + ). x + - x + + x x + ( x + ) ( x + ) ( x ) ( x + ) - ( x + ) ( x + ) + ( x + ) ( x + ) ( x + ) ( x + ) + ( x ) ( x + ) Express as a single fraction. - ( x + ) ( x + ) Simplify the numerator by expaning brackets an collecting like terms. Note: The enominator is generally kept in factorise form. That is, it is not expane. ( x + x + x + ) + ( x x + 6x ) ( x + ) ( x + ) x + x + + x + x ( x + ) ( x + ) x + 8x - ( x + ) ( x + )

14 Maths Quest 0 for Victoria remember remember. Algebraic fractions contain pronumerals that may represent particular numbers or changing values.. To a or subtract algebraic fractions we perform the following steps. (a) Fin the lowest common enominator (LCD) by fining the lowest common multiple (LCM) of the enominators. (b) Rewrite each fraction as an equivalent fraction with this common enominator. (c) Express as a single fraction. () Simplify the numerator. E Aing an subtracting algebraic fractions SkillSHEET.7 8 Simplify the following expressions. y y y y a - b c 8 x x - SkillSHEET.8 8x x w w e f g y y 0x x + h i y y - 0 x + x x + x + 6 x x + j k l 6 x + x + + Mathca Aing an subtracting algebraic fractions 9 Simplify the following. a b - - x 8x x x c - + e x x 6x 8x f g h + 00x 0x 0x x i x 7x x x - - x x Simplify the following by writing as single fractions. 0 a b ( x + ) x ( x x ) ( x + ) ( x ) c x - e ( x + ) ( x 7) ( x + ) x ( x ) f g x + - h x + x x + - x + x + x 7 x + 6 i j x + x x k x + x 8x + 9 7x x + l x - ( x + ) + ( - x ) x x ( x + 7) + ( - x ) x + 8 x + - x + x + x + x - x + x

15 Chapter Algebra an equations Multiplying an iviing algebraic fractions The rules for multiplication an ivision are the same as for numerical fractions. When multiplying algebraic fractions, multiply the numerators an multiply the enominators, then cancel any common factors if possible. x xy x xy For example, - y 7 y 7 8x - (Cancel y from the numerator an enominator.) When iviing algebraic fractions, change the ivision sign to a multiplication sign an write the following fraction as its reciprocal (turn the fraction upsie own). 8x x 8x For example, The process then follows that for multiplication. x Simplify each of the following. a y 6z - - x 7 y b x x ( x + ) ( x ) x a Write the expression. a Multiply the numerators an multiply the enominators. Check for common factors in the numerator an enominator, an cancel. (0 an have a common factor of. Cancel y.) b Write the expression. b Multiply the numerators an multiply the enominators. Check for common factors in the numerator an enominator, an cancel. Cancel x. Cancel (x + ). y 6z - - x 7y 0yz - xy 0z - 7x x x + - ( x + ) ( x ) x x( x+ ) - xx ( + ) ( x ) x

16 Maths Quest 0 for Victoria Simplify the following expressions. a b xy x - 9 y x 7 - ( x + ) ( x ) - x + a Write the expression. a Change the ivision sign to a multiplication sign an write the secon fraction as its reciprocal. Multiply the numerators an multiply the enominators. Check for common factors in the numerator an enominator an cancel. Cancel x. xy x - 9y xy 9y - x 7xy 8x 7y 8 b Write the expression. b Change the ivision sign to a multiplication sign an write the secon fraction as its reciprocal. Multiply the numerators an multiply the enominators. Check for common factors in the numerator an enominator an cancel. Cancel (x + ). ( x + ) ( x ) - x 7 x + ( x + ) ( x ) - x + x 7 ( x + ) - ( x + ) ( x ) ( x 7) ( x ) ( x 7) remember. When multiplying algebraic fractions, multiply the numerators an multiply the enominators an cancel where possible.. When iviing algebraic fractions, change the ivision sign to a multiplication sign an write the following fraction as its reciprocal (turn the fraction upsie own). remember

17 Chapter Algebra an equations F Multiplying an iviing algebraic fractions a b Simplify each of the following. x 0 a - y y 6 c - x x e y y 8z g - - x 7y x 9z i - z y 0y z k - 7x y Simplify the following expressions. x x a - ( x ) ( x ) x c e 9x x + ( x + ) ( x 6) x x x - x + - ( x + ) ( x ) b f h j l b f x - y x 9 - y w 7 x y 6z - - x 7y y x - - x 8y y x - w y x x + 7 ( x ) ( x + 7) x ( x + ) x + - ( x + ) ( x + ) - x + xx ( + ) - x( x ) SkillSHEET.9 SkillSHEET.0 Multiplying an iviing algebraic fractions Mathca a Simplify the following expressions. a x x c - x x e - w w xy x g - 7 y 6y x i - 9 xy xy xy k b f h j l 9 x x y y x x xy x - y 8wx w - y 0xy 0x - 7 y SkillSHEET. Simplify the following expressions. b 9 x + a ( x ) ( x 7) - x c e x x + - ( x + ) ( x ) x + ( x ) ( x ) - ( x + ) ( x 7) x 7 b f x 9 ( x + ) ( x ) x ( x + ) x + - ( x + ) ( x ) - x x 7 ( x 8) ( x ) - ( x 8) WorkSHEET.

18 6 Maths Quest 0 for Victoria Solving basic equations Equations are algebraic sentences that can be solve to give a numerical solution. This section covers basic one-step an two-step equations as well as equations containing brackets. Remember that to solve any equation we nee to uno all the operations that have been performe on the pronumeral. Solve the following equations. a a b - 6 c e 0.87 f a Write the equation. a a Subtract 7 from both sies to obtain the solution. a a b Write the equation. b Express as an improper fraction Multiply both sies by 6 to obtain c Write the equation. c 0.87 e Square both sies to fin e. ( ) 0.87 e Write the equation. f Take the square root of both sies to obtain f. Note that there are two possible solutions, one positive an one negative. f ± f ± e Each of the equations in worke example was a one-step equation. Remember that in two-step equations the reverse orer of operations must be applie.

19 Chapter Algebra an equations 7 Solve the following. a y 6 79 b x - 9 a Write the equation. a y 6 79 A 6 to both sies. y y 8 Divie both sies by to obtain y. y y 7 x b Write the equation. b - 9 x Multiply both sies by x Divie both sies by to obtain x. x - - x - x Graphics Calculator tip! Solving equations Your graphics calculator has the facility to solve equations. To solve an equation the x equation shoul be written in a form equal to 0. Consier the equation - in 9 x worke example (b). The equation will nee to be written as Press the MATH button an choose option 0: Solver. You shoul obtain the screen shown.

20 8 Maths Quest 0 for Victoria If this is not isplaye, there must be a previous equation store in the memory. Press until this screen is isplaye an clear any equation that has been store. x. Enter the expression - by pressing ALPHA [X] 9 an press ENTER. 9. Press ALPHA [SOLVE] to obtain the solution for X. Equations where the pronumeral appears on both sies We can also solve equations where the pronumeral appears on both sies of the equation. We aim to a or subtract one of the pronumeral terms so that it is eliminate from one sie of the equation, usually the right-han sie. Solve the following equations. a h + h b 7 a Write the equation. a h + h Subtract h from both sies. h + Subtract from both sies. h Divie both sies by. h b Write the equation. b 7 A to both sies. 7 Subtract from both sies. Divie both sies by. - remember. Equations are algebraic sentences that can be solve to give a numerical solution. remember. Equations are solve by unoing any operation that has been performe on the pronumeral.

21 Chapter Algebra an equations 9 G Solving basic equations a b c, a b Solve the following equations. a a b k 7 6 c g r. 0.7 e h f i + g t 7 h q + i x Solve the following equations. f i a b - 6 c 6z 0 9v 6 e 6w f m 7 y g a.7 h - i Solve the following equations. a t 0 b y 89 c q. f. e h f p g g - h j i a Solve the following. a a b 6b + 8 c 8i 9 7f 8 e 8q f 0r g 6s h t 8 i 8a Solve the following. f g r a b + 9 c m n p - 0 e + 8. f Solve the following. a 6(x + 8) 6 b 7(y ) c (m ) 7 (k + ) e (n ) 80 f 6(c + 7) 8 7 Solve the following. k 9m 7 p a - b 8 c u x v - e f k EXCEL Equation solver: ax + b c Solving basic equations Spreasheet GC Solving linear equations EXCEL Equation solver: a(bx + c) Mathca program Spreasheet

22 60 Maths Quest 0 for Victoria 8 multiple choice p a The solution to the equation + 7 is: A p B p C p D p 0 E p b c If h + 8, then h is equal to: A B. C D 0 E 9 The exact solution to the equation x 7 is: A x.7 87 B x.7 (to ecimal places) C x D x. E x. - EXCEL a Equation solver: ax + b cx + Spreasheet b 9 Solve the following equations. a x b c p 7 x e h 0 f 6t 0 g v r h - i g. 0 Solve the following equations. a 6 x 8 b 0 v 7 c 9 6l g e t 7 f e - g 8 j k f 9 h 6 i Solve the following equations. a 6x + x + 7 b 7b + 9 6b + c w + 7 6w + 7 8f 7f + e 0t t + f r 6 r + g g 9 g h 7h + h 6 i a a Solve the following equations. a x 6 x b 0 c 8 c c r + 9r k k 6 e y + 8 y + 7 f 7 g g g w w + 8 h m m i p 9 p Solve the following equations. a (x + ) x b 8(y + ) y c 6(t ) (t + ) 0(u + ) (u ) e ( f 0) ( f ) f (r + ) (r + 7) g ( + 9) ( + ) h (h ) (h ) i (x + ) ( x) GAMEtime Algebra an equations 00 multiple choice a The solution to 8 k is: A k B k C k D k E k 6n b The solution to is: A n B n C n D n 8 E n 8 c The solution to p 6 8 p is: A p B p C p D p E p

23 Chapter Algebra an equations 6 Musical notes You shoul be quite familiar with Pythagoras theorem involving the sies of a right-angle triangle. Di you know that Pythagoras mae many other important mathematical iscoveries? One such iscovery was to o with musical notes. fourth fourth G C F Pythagoras note that if string lengths are in the ratio :, fourth fourth the strings prouce notes that are four notes apart. G an C are four notes apart. A string that prouces note G is cm long. Write an equation that will enable you to fin the length of the string for note C. Note F is four notes above note C an is prouce by a G A B C DEF string that is 7 cm long. Write an equation that you can String lengths solve to fin the length of the string for note C. If the string for note G is g cm long, write a general formula for the length of the string for note C (c cm). If the string for note F is f cm long, write a general formula for the length of the string for note C (c cm). For a particular piano, the note C is prouce by a string that is 70 cm long. Use your formula from part to fin the length of the string neee to prouce the note G below note C. 6 Use your formula from part to fin the length of the string neee to prouce the note F above the note C. Factorise 6 6y. Factorise + 8g. 6 Simplify - -. x x x( x + ) Simplify. x - ( x + ) ( x ) x ( + ) Solve the equation r 6. t 6 Solve the equation Solve the equation 8( + v) 0. y 8 Solve the equation Solve the equation z z 9. 0 Solve the equation 9(b ) 7(0 + b).

24 6 Maths Quest 0 for Victoria MATHS MATHS QUEST H C A L LL E N G G E E I think of a number. If is ae to this number an then quaruple, my answer is. What number i I think of? At a special meeting, each person shakes hans exactly once with every other person. If there are 6 hanshakes altogether, how many people attene the meeting? When Simon was born, his father was 6 years ol. Five years ago, his father was twice as ol as Simon will be in 7 years time. How ol is Simon? Solving more complex equations Now that we have reviewe some algebraic skills, let s apply them to some more complex equations. Equations with multiple brackets Many equations nee to be simplifie by expaning brackets an collecting like terms before they are solve. Doing this reuces the equation to one of the basic types covere in the previous exercise. Solve each of the following linear equations. a 6(x + ) (x ) 0 b 7( x) (x + ) 0 6 a Write the equation. a 6(x + ) (x ) 0 Expan all the brackets (Be careful 6x + 6 x with the ). Collect like terms. x + 0 Subtract from both sies of the x equation. Divie both sies by to fin the value of x. x 7 b Write the equation. b 7( x) (x + ) 0 Expan all the brackets. 7x x + 0 Collect like terms. 7x x 7 Subtract x from both sies. 0x 7 Subtract from both sies. 0x 6 Divie both sies by 0 to solve x - for x. 0 x

25 Chapter Algebra an equations 6 Equations with algebraic fractions To solve an equation that has algebraic fractions, we rewrite the equation as equivalent fractions using a common enominator. Multiplying both sies of the equation by this common enominator removes the fraction/s from the equation. Again, this shoul reuce the equation to one of the more basic equations. Solve each of the following linear equations. x x a - b - x x x a Write the equation. a Simplify the left-han sie of the equation by subtracting the algebraic fractions using equivalent fractions with a LCD of 0. Multiply both sies of the equation by the common enominator of 0. Divie both sies by to obtain x. b Write the equation. b 7 Collect the algebraic fractions on one sie of the equation by aing x to both sies. 0 x x x 8 Simplify by aing the algebraic x x fractions. Use x as the LCD to form equivalent fractions. - x Multiply both sies of the equation - x x x by the LCD of x. x x Divie both sies by to obtain x. - - x or x An alternative metho of solving equations with algebraic fractions is to write every term in the equation as a fraction with the same common enominator. Every term can then be multiplie by this common enominator. This has the effect of eliminating the fraction from the equation. x - 0 x - 6x - 0 x - 0 x x x - + x x

26 6 Maths Quest 0 for Victoria 8 Solve each of the following linear equations. ( x + ) ( x ) a b - 6 ( x ) - x + a Write the equation. a ( x + ) ( x ) ( x + ) 0 8( x ) Write each term as an equivalent fraction with a common enominator of 0. Multiply each term by 0. This (x + ) 0 + 8(x ) effectively removes the enominator. Expan the brackets an collect like terms. x x 8 x x Subtract 8x from both sies. 7x Subtract 7 from both sies. 7x 7 Divie both sies by 7 to obtain x. x 7 x b Write the equation. b - - ( x ) x + ( x + ) ( x ) Write each term as an equivalent ( x ) ( x + ) ( x ) ( x + ) fraction with a common enominator of (x )(x + ). Multiply each term by the common (x + ) (x ) enominator. Expan the brackets. x + x Subtract x from each sie of the x + 6 equation. Subtract from each sie to obtain x. remember remember x + x 7. For more complicate equations with brackets, expan the brackets an collect like terms. This will reuce the equation to a more basic type.. For complicate algebraic fraction equations, follow either of the two techniques: (i) Simplify by aing or subtracting the algebraic fractions. (ii) Write each term in the equation as an equivalent fraction with a common enominator. Then multiply each term by the common enominator, to remove the fraction from the equation.

27 Chapter Algebra an equations 6 H Solving more complex equations 6 Solve each of the following linear equations. a 6(x ) + 7(x + ) 9 b 9( x) + (x + ) 0 c 8( x) ( + x) 9( + x) 8(x + ) x e 6( + x) 7(x ) + f 0(x + ) (8 x) + 6 g 8(x + ) + (x ) 6(x + ) h 6(x ) (6 x) 7(x ) i 7.(x ) +.( x). j (x ) (6x + ) 8 k 9(x ) + (6x + ) 00 l 7(x + 7) (x + ) ( x) Solving more complex equations Mathca 7 Solve each of the following linear equations. a x x x x + - b c x x x x + e f g x x - h x 6 x i x x j + k + 6 l x x x x - 7 x x x - x x x + 0 GC Solving linear equations program Solve each of the following linear equations. 8 ( x + ) ( x + ) a b c e g i k x ( + ) 6( x ) - 7 ( x) x ( ) ( x ) 6x ( ) - x + - x + - x x - x 8 - x + f h j l ( x + ) x ( ) ( x + ) - ( x + ) - 6 ( x) 9( x + ) x ( ) 7 - x x x x ( x ) - - x - x + multiple choice a Which of the following is the solution to the linear equation (x ) (x + ) 0? A x. B x. C x 0.9 D x 0.9 E x. x x b Which of the following is the solution to the linear equation -? 7 7 A x B x C x D x E x 0.6

28 66 Maths Quest 0 for Victoria Solving inequations Inequations involve the inequality signs > (greater than), (greater than or equal to), < (less than) an (less than or equal to). In most cases, we can solve an inequation as if the inequality sign was an equals sign. Solve the following inequations. a x + > 7 b c 7 7 a Write the inequation. a x + > 7 Subtract from both sies. x > b Write the inequation. b c 7 7 A 7 to both sies. c Divie both sies by to obtain c. c - 9 c n Solve - < 6. 7 If we multiply or ivie both sies of an inequation by a negative number, the inequality sign must be reverse. To see why this must be one, consier the true inequality statement 6. When we multiply or ivie both sies by, we obtain 6. Clearly, this is no longer true because is not less than 6. Reversing the inequality sign corrects the statement so that it becomes 6. When multiplying or iviing an inequation by a negative number, the inequality sign nees to be reverse. 0 Write the inequation. n - < 6 7 Multiply both sies by 7. n < Divie both sies by an reverse the inequality sign as we are iviing by a negative number. n > The solution to an inequation may be graphe on a number line. This is one by placing a circle on or above the number which solves the matching equation together with an arrow in the irection of the inequality. For < or >, the circle is hollow or open, while for or, the circle is fille in or close.

29 Chapter Algebra an equations 67 Consier an inequation with solution x. The circle is rawn at an fille in with the arrow pointing left as shown below x Solve the following inequation an sketch the solution on a number line. 7h + > h 7 Write the inequation. 7h + > h 7 Subtract from both sies. 7h > h Subtract h from both sies. h > Divie both sies by. h > Sketch the solution. Use an open circle since > oes not inclue. remember remember h > - or h. Inequations involve the inequality signs > (greater than), (greater than or equal to), < (less than) an (less than or equal to).. In most cases we can solve an inequation as if the inequality sign was an equals sign.. If we multiply or ivie both sies of an inequation by a negative number, the inequality sign must be reverse. I Solving inequations 9a Solve the following inequations. a x + > 6 b v 8 < c t + 7 p e f + 7 f h > g k +.6 <. h j i y + > Solve the following. r a 6x < 8 b c y > e k e > 7 f > 8 EXCEL Solving linear inequations Solving linear inequations Spreasheet Mathca

30 68 Maths Quest 0 for Victoria 9b 0 Solve the following inequations. a 9h + 8 < b 8 c y + 8u c 6p > e - 9 f - > y k x g + h 7 < i Solve the following inequations. a x < b 6 h > c 8u k 9 j x e - < f 6 f w a g + > 9 h i - > 7 8 Solve the following inequations an sketch their solution on separate number lines. a i > b u + 7 < 8 c 7g < 6 c m > e 8y + 7 f - + > 8 g ( j + ) 9 h 6x + x + 7 i r > 8r j 7 w < 8 + w k (x + ) > 7x l 6( 8 y) ( 7 y) WorkSHEET. 6 multiple choice a The solution to x + 6 < 8 is: A x < - B x > - C x < D x > E x > b c The solution to the inequation x is: A x B x C x D x E x The solution to the inequation x + > 6x + is: A x < B x < C x > D x > E x < 9 Dance fever At the beginning of the chapter, we consiere the case of a school ance, from which the organisers hope to raise at least $000 for charity. The tickets to the ance cost $6 each with costs of $00 for hire of the venue an $0 for the DJ. Write an inequation to represent this situation. Explain what your chosen pronumeral represents. Solve the inequation to fin the number of people that must atten the ance to raise at least $000. Fin the number of people attening the ance if $00 profit is mae. If the organisers hire a ban instea of a DJ an the cost of the ban is $900, fin the minimum cost of a ticket to maintain a $00 profit if the same number of people atten.

31 Chapter Algebra an equations 69 What is a black hole? Solve the inequations to fin the puzzle answer coe. S x ( x) x > x L T x + < 7 (x ) x 7(x ) 6x A E N U x > x 7 x x 7 < x > B G O V 6 x < 0 7x x > x (x + ) C H P W x 7 + x > 6 x 7 + > 0.x + > 8 D I R Y x < x > x x 0 x x x < x > 6 x < x 0 x > x < 7 x x > 0 x x x < 7 x x x 9 x > x < x < x > 6 x < x > 0 x > 6 x x x < 7 x x < x > x < 7 x 9 x 9 x < x x < x < x 0 x > 6 x 9 x x < x > x < x 9 x < x x x > x < x < x x > 6 x < x x < x x 0 x 0

32 70 Maths Quest 0 for Victoria summary Copy the sentences below. Fill in the gaps by choosing the correct wor or expression from the wor list that follows. Like terms contain the same parts an can be collecte (ae or subtracte) in orer to simplify an algebraic expression. When an iviing algebraic terms it is not necessary to have like terms. When the numerical values of pronumerals are known, we can them into an algebraic expression an evaluate it. It is sometimes useful to place substitute values in when evaluating an expression. brackets in an algebraic expression is achieve by multiplying the term outsie the brackets by each of the terms insie. 6 To factorise an expression: Fin the common factor (HCF) of the terms an place in front of the brackets. each term by the HCF an place insie the brackets. 7 To a or subtract algebraic fractions we perform the following steps. Fin the common enominator (LCD) by fining the lowest common multiple (LCM) of the enominators. Rewrite each fraction as an equivalent fraction with this enominator. Express as a fraction. Simplify the numerator. 8 When multiplying algebraic fractions, multiply the together, an multiply the enominators together. 9 When iviing algebraic fractions, change the ivision sign to a multiplication sign an write the following fraction as its (turn the fraction upsie own). 0 Equations are algebraic sentences that can be solve to give a solution. To solve an equation we o the same to both sies of the equation in orer to the pronumeral. Inequations involve the signs > (greater than), (greater than or equal to), < (less than) an (less than or equal to). If we multiply or ivie both sies of an inequation by a number, the inequality sign must be reverse. WORD LIST lowest reciprocal brackets multiplying isolate numerical common pronumeral single substitute negative expaning ivie numerators highest inequality

33 Chapter Algebra an equations 7 CHAPTER review Simplify the following by collecting like terms. a c + c 8 b k + m k 9m c + c 8c 6y + y + y 7y Simplify the following. abc 8mc a m 7p b - c - 80bc 0mf multiple choice The expression 6 + r r simplifies to: A + r B 0 + r C 0 r D + r E 8r If A bh, fin the value of A if b 0 an h 7. multiple choice 0a b ab Given E mv an m 0., v 0. then E A B 0. C 0.00 D 0.0 E Expan the following. a (x 6) b ( x) A A A B B C 7 multiple choice The expression (f + ) + 6(f 7) simplifies to: A f + B f C f 7 D f + E 6f C 8 multiple choice The expression 7(b ) (8 b) simplifies to: A 8b 9 B 8b C 6b 9 D 6b E 8b + 9 Factorise the following. a ap ag b h 7 c p 6 + p C D 0 multiple choice The expression a y 0ay can be fully factorise as: A ay(y a) B ay(a y) C a y (y a) D ay(y.a) E ay(a.y) D Simplify the following. y y x + x + a - b c - - x x x + x - x + + x + E

34 7 Maths Quest 0 for Victoria F Simplify the following. a y 0y z - b x 7x 6y c 0 xy 0x - - e - x x y f x + 6 ( x + ) - ( x + ) ( x + ) - x + 6 x 9x + - ( x + 8) ( x ) x + 8 G G H A H I test CHAPTER I yourself Solve the following equations. a p 0 68 b s 0.6. c b 8 r e x f (x + ) 7 y g h a 6 i k 7 Solve the following. a 7b b t t + c (p ) (p ) Solve each of the following linear equations. a (x ) + (x + ) 0 b 7( x) ( x) c (x + ) 6(x ) 7(x + ) 8(x ) + (x ) 7x e 7(x ) (x + 0) x f (x + ) + 6(x + ) x Solve each of the following equations. x x x x a + b c x x + + e - f x x 7 Solve the following inequations. a x + > 7 b x < c e y e 8u f x 8 x Show the solution to the inequation n > 0 on a number line. - x x 7 6 ( x + ) ( x + )