Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity 9:0 Equtions with pronumerls on both sides Syllbus references (See pge for detils.) 9:0 Equtions with grouping symbols GeoGebr ctivity 9:0 Equtions with grouping symbols 9:0 Solving simple qudrtic equtions 9:06 Solving problems using equtions Mths terms, Dignostic test, Assignments Number nd Algebr Selections from Equtions [Stge ] Solve simple liner equtions. (ACMNA179) Solve liner equtions using lgebric techniques nd verify solutions by substitution. (ACMNA19) Solve simple qudrtic equtions. Working Mthemticlly Communicting Problem Solving Resoning Understnding Fluency 0
In Yer 7 you were introduced to equtions. An eqution is number sentence in which one (or more) of the numbers is unknown nd is represented by pronumerl, usully letter of the lphbet. e.g. + = 1 or 10 = We solve n eqution to find the vlue of the pronumerl. To solve equtions it is necessry to understnd inverse opertions. 9:01 Inverse opertions Prep quiz 9:01 Write n lgebric epression for ech of the following sentences. 1 Add 6 to. Multiply by 8. Divide y by 7. Tke wy from p. Multiply by nd then dd. 6 Subtrct from y nd then multiply the result by 9. Wht is the opposite of: 7 dding 7 8 subtrcting 8 9 multiplying by 9 10 dividing by 10? We strt with pronumerl when building lgebric epressions. Algebric epression Mening Arrow digrm + hs dded to it. + + hs subtrcted from it. is multiplied by. is divided by. Arrow digrms cn be used to show how the epression hs been mde. The opertion tht hs cused the chnge is written bove the rrow. Equtions re sometimes clled lgebric sentences. 9 Equtions 1
When solving equtions, we need to reverse the steps to get bck to the pronumerl. In other words, we need to perform the inverse opertions. + + then + - then - + then then Inverse mens opposite. + 7 7 Adding nd subtrcting re inverse opertions. This process is clled bcktrcking. Worked emples 1 Complete the following rrow digrms by writing the correct opertion bove ech rrow. + 6 c - ( - ) b y y y + 7 d - 8 8 9 Use the inverse opertions to get bck to the pronumerl. 6 + 6 c ( - ) + Solutions b d y + 7 7 8 9 9 +8 1 +6 + 6 b y y +7 y + 7 c - ( - ) d 8-8 9 8 9 Order of opertions is very importnt. + 6 6 b y + 7 7 y y c ( - ) - + d 8 9 9-8 +8 Austrlin Signpost Mthemtics New South Wles 8
Eercise 9:01 Foundtion worksheet 9:01 Algebr review 1 Complete the following rrow digrms by writing the correct opertion bove ech rrow. + 6 b m m - c n 7n d e y y + 1 (y + 1) f p p - 6 9(p - 6) g + 1 + 1 t h t t 7 7 10 i q q m m k m - 8 6 6 l - q + j + 8 Drw rrow digrms to show how the following epressions were mde from ech pronumerl. 9 b y + c q 9 d e n + 6 f 7m 1 g p + h 9y i + j + n k + 7 l 1 m (n + ) n 8( 1) o (q + 9) p 10(t 1) q 6 r 7v + 1 s 7 t 9 6 Write the opertions tht must be performed on to produce the following epressions. 7 b 7 c + 1 d 8 e + f 7 + g 9 h 6 + i 7 j 10 + k 7 These flowchrts show how the epressions re mde l + 1 m ( + 7) n 9( ) o -7 + 1 p 1 9 Equtions
Complete the following rrow digrms. b c m d t +9 e p + f 6 g y +1 7 h b + i q j n ( ) +6 k b + 10 10 l - 7 +7 m m o + q 6 + 7t s n n p p - + 6 7 r m + + t - ( ) Complete these rrow digrms, showing how to get bck to the pronumerl. m m b c n + n d p - 7 p e 10 f g 8 + t t h q - q i m + 1 m m j p - p p k 7 + 7 l 9q - 9q q Bcktrck to the pronumerl. m 6 + n 7 + m m m o p + 7 + 7 q b b b r s 8 p p p t Austrlin Signpost Mthemtics New South Wles 8
6 Wht opertions must be performed on these epressions to bcktrck to? b c + 9 d 8 e + 1 f 7 + g 8 - h 6 + i j 1 + k 7 10 l + 1 m 8( + 7) n ( 11) o - + 1 p 7 Wht order of inverse opertions must be performed to get bck to the pronumerl? 9( ) b (8y + 7) c ( + 9q) d + e n f + g 7 + 9 1 h 7 i ( 6) j 7(v + 1) k l 9 6 9:0 Solving equtions Prep quiz 9:0 Simplify ech of the following. 1 7 7 + 7 + 7 m Wht is the inverse of: multiplying by 6 dding 7 7 subtrcting 1 8 dividing by? Complete these rrowing digrms, putting opertions bove ech rrow. 9 m + m m 10 8-8 Solving equtions is like blncing scles. With equtions, we know tht one side is equl to the other. The solution of the eqution therefore is the vlue of the pronumerl tht blnces the eqution. + 7 = 10 = 0 + 7 + 7 10 10 6 0 0 = blnces the scle so = is the solution. = 6 blnces the scle so = 6 is the solution. 9 Equtions
Often, solving n eqution requires us to chnge the eqution into simpler one. To do this we dd (+), subtrct ( ), multiply ( ) or divide ( ) both sides of the eqution by the sme number. Look t the solutions of these two equtions, noting tht both sides remin blnced becuse the sme opertion is done to both sides. + = 1 We need to perform opertions tht will leve only the pronumerl on one side of the eqution. + 1 1 Solve ech of these one-step equtions. Cn you see why it s importnt to know bout inverse opertions nd how to work bck to the pronumerl? + Scles? = is the solution Note here tht the inverse opertions re: 1 Worked emple 1 1 +1 1 1 1 = 1 + 1 + 1 16 = 16 is the solution m + 7 = 9-7 - 7 m = 9 7 m = b 16 = 9 + 16 + 16 = 9 + 16 = c p = p = p = 7 d = 1 7 7 7 = 7 1 = 8 6 Austrlin Signpost Mthemtics New South Wles 8
Worked emple Solve ech of these two-step equtions. + + = 7 b = - - = + = 10 - - = 1 = 7 c p 6 = 1 + 6 + 6 p = 1 p = 6 d 10 m = - 10-10 -m = 1 - - m = Eercise 9:0 Foundtion worksheet 9:0 Solving equtions 1 1 Solve ech of these one-step equtions. + = 1 b q + 9 = 1 c y + 1 = 1 d 9 + n = 1 e 8 + p = 1 f 0 + t = g 7 = h w 8 = 8 i m = 11 j = 0 k 7y = l y = 8 m = n = m o = 6 6 Solve these one-step equtions tht involve negtive integers. + = 1 b m + = c + 1 = 8 d + n = e + p = -1 f 7 + y = 0 g n 7 = - h t 8 = - i q = j = -1 k 7y = -1 l = m - = 8 n -8y = - o -6 = p = w q = -1 r v s ( 7) = 1 t t ( 8) = 10 u q ( ) = Substitute the solution into ech eqution below nd stte whether the solution is correct or not. + 17 = 9 b 9 + m = 1 c q 1 = 16 d n 8 = 17 = 6 m = 7 q = 9 n = 11 e 17 d = 9 f 9 p = 1 g y = 9 h n = d = 8 p = y = 7 n = 7 i n = -0 j = 8 k = n l = 7 7 n = = k = 8 n = 1 9 Equtions 7
Solve these two-step equtions. Clerly show ech step in your working. (All the nswers re integers.) + = 7 b n + = 8 c y + 1 = 16 d n = 9 e 6k = 7 f 9q = g w = 11 h + p = i 1 + 7s = 6 j 1 = 6 k 16 j = 6 l 0 9g = m 1 = v + 6 n 0 = 8 + y o 10 = p 6 = - q p = - r 1 = 6 h Solve these two-step equtions. (The solutions re ll positive integers.) t = b = 6 c = + 1 + m + 8 d = e = f = n 1 y 7 g = h = i = 8 h j + 1 = k + = 7 l 1 = m + = 9 n + q = 8 o 9 n = 6 Find the solutions to ech of the following equtions. + = b n + 6 = 8 c y + 1 = d n 1 = 9 e 6k = 7 f 6q = g + w = 10 h + p = i 1 + 7s = 0 j 1 = 8 k 1 j = 6 l 0 7g = m 1 = v + n 1 = 8 + y o 10 = y p 6 = - q p = r 1 = 10 h 7 Solve this set of equtions, which re either one- or two-step types. = 7 b n + = 8 c y + = 1 d y = 6 e 7 + = 17 f y = - g y + = 1 h y 1 = 1 i = 0 j + m = 8 k y = 9 l y = 9 t m = n = 9 o = q p 1 = w q + 1 = 6 r = y + 7 6 s = t = u 11 = 7 t + p v + = 11 w = = 8 = 6 = 0 = 10 Multiply ech term on both sides by the denomintor. + = 6 = = or 11 Opposite opertions re the key. 8 Austrlin Signpost Mthemtics New South Wles 8
Fun spot 9:0 Work out the nswer to ech prt nd write the letter for tht prt in the bo bove the correct nswer. Solve these equtions. T + 8 = 1 A = 6 A = ( 1) T + = 1 T 8m = 6 D 11 = 11 A + = 1 T 7 = 1 E 0 = -1 E 8 = 0 S 8 + = 1 H 1 = U 60 n = 0 O n = 1 N ( 9) = 0 T n = 0 n U 7 = 8 O 100 m = O 6 = W 1 + n = 1 T 7 = 7 I - = K 1 = -1 S = 1 B - = 8 T 8 = I = m T 10 n = - E = 0 0 11 8 100 0 9 7 WHY DID THE tooth GET DRESSED UP? 7 1 6 6 10 1 10 1 8 How hevy is one bo? 1 8 1 60 1 9 Equtions 9
Mths terms 9 lgebric epression n lgebric epression consists of one or more terms joined together by opertion signs, e.g. +, +, m 1 7 n epression does not hve n equls sign like n eqution eqution number sentence where one or more of the numbers is missing or unknown the unknown number is represented by pronumerl, e.g. + = 8, + 1 = 7 evlute to find the vlue of n epression, e.g. Find the vlue of + 7 if = to find the nswer (usully fter substitution) grouping symbols the most common types re: prentheses ( ) brckets [ ] brces { } used to group number of terms together in n epression, e.g. ( + ) inverse opertion the opertion tht will reverse or undo previous opertion, e.g. Addition is the inverse opertion of subtrction; division is the inverse opertion of multipliction pronumerl symbol used to represent number usully letter such s qudrtic eqution qudrtic equtions hve squred pronumerl term nd usully hve two solutions e.g. = 9 hs two solutions: = 7 or = -7 solution the nswer to problem OR the method of finding the nswer to problem the solution to n eqution is the number tht mkes it true sentence e.g. = is the solution to + = solve find the solution or nswer to problem or eqution substitute to replce pronumerl with numerl e.g. To substitute for in the epression would give: () = 1 = 10 70 Austrlin Signpost Mthemtics New South Wles 8
DIAGNOSTIC TEST 9 Equtions Ech section of the test hs similr items tht test certin type of question. Errors mde will indicte res of wekness. Ech wekness should be treted by going bck to the section listed. 1 Solve: + 11 = 7 b p = c + = d y 7 = - Solve: 9:0 n 6m = b = c = 8 d = Solve these two-step equtions. (Answers re integers.) + = 7 b m 1 = 8 c 6 + y = d = 6 The nswers to these equtions involve frctions. + 1 = 6 b y = c 7 + k = d 9 t = 6 Solve these two-step equtions. 9:0 m m + w = 10 b = c + = 6 d = 6 Solve: + 1 = + 7 b m + = m 6 c k 1 = k d 7 = + 9 7 Solve: ( + ) = 8 b 6(m ) = 1 c (y + 1) = 6 d (m 1) = 7 8 Solve: (n + ) = n + 7 b 9p + 1 = 7(p + ) c ( + 1) = ( ) d ( 1) = ( 1) 9 Solve: = 16 b c = 9 c m = 100 d b = 10 Form n eqution for ech number problem, letting the number be n, nd solve it. The product of number nd 7 is 6. Find the number. b The sum of number nd is. Wht is the number? 11 Form n eqution from the informtion with ech shpe, nd solve it to find. b c ( + ) cm 9:0 9:0 9:0 9:0 9:0 9:0 9:0 9:06 9:06 6 cm cm Perimeter = 80 cm cm Are = 60 cm ( + 0) 8 9 Equtions 71
ASSIGNMENT 9A Chpter review 1 Complete these rrow digrms by filling in the boes, showing the bcktrcking steps. ( - ) + b 9y + 7 7 9 c 8 +8 Solve these simple one-step equtions + 1 = 1 b m = c 1 n = d t 9 = 9 e 6y = - f = 1 g 8 = 8 h -y = 10 i w = 6 j = 8 k (-) = 6 l t ( 8) = - m = 6 n c = 81 o m = 7 p b = 11 Solve these two-step equtions. + = b n + = 8 c n = 17 d k = 7 e 9q = 1 f + w = 7 g + 9p = h 1 + s = 6 i 1 = 7 j 16 j = 6 k 0-8g = l 1 = 9v + 6 m 7 = + y n 1 = o 6 = 6 p p = n y + 7 q = r = 1 6 h s 7 = t + 7 = Solve these equtions tht hve the pronumerl on both sides of the eqution. + = + 7 b y = y + c 7w + 1 = 9w d + p = p + 8 e = + 9 f 7k + = 9 k Solve: ( + ) = 6 b (n + ) = 1 c (n 1) = 0 d (k ) = e (q + ) = 10 f ( + w) = 16 g 6( ) = 9 h ( + ) = 9 6 Solve these equtions. ( + ) = + 8 b + 9 = 6( + 1) c (y ) = (y + ) d (1 + p) = (p + 8) e ( + q) = (7 q) f ( ) = ( + 9) 7 Write n eqution for ech problem nd solve it. Let the number be. When is dded to the product of number nd 7 the nswer is 61. Find the number. b If 9 is subtrcted from number nd the result is multiplied by 7 the nswer is -1. Find the number. 8 Form n eqution nd solve it to find. 6 cm (8 + ) cm Are = 70 cm 7 Austrlin Signpost Mthemtics New South Wles 8
ASSIGNMENT 9B Working mthemticlly 1 I purchsed 1 boes of mtches. If ech bo contins between 7 nd mtches, choose which of the following nswers could be the totl number of mtches. A 61 B 70 C 76 D 810 Sophi is 8 yers younger thn Alec. Romn is one-qurter the ge of Sophi. Mri is 18 yers older thn Romn. If Mri is 8, how old is Alec? One cn of food hs mss of 0 g, wheres nother hs mss of 0 g. A third cn hs mss of 67 g nd fourth cn hs the second lrgest mss of these four cns. The totl mss of the four cns is: A 167 g B 18 g C 186 g D 1 g To celebrte the birth of new ntion, 00 people link hnds to rech cross the city of Sydney. If the spce between bodies is 1 m nd the width of ech body is 0 m, how fr will the people stretch? The totl number of votes cst in three electortes ws 68 9. If in one of these electortes 9 06 votes were cst nd in nother 198 86 votes were cst, how mny votes were cst in the third electorte? 6 When three cubes re joined t their fces they cn only form two different solids. These re shown here. How mny different solids cn be formed by joining four cubes? Some of the solids mentioned in Question 6 bove cn be interlocked to form this fmous puzzle clled the Som Cube. Find out which ones they re. 9 Equtions 7
ASSIGNMENT 9C Cumultive revision 1 Before they plyed, Sndy hd 1 mrbles nd Luke hd 10. Luke lost mrbles to Sndy. Wht percentge of his mrbles did Luke lose? b By wht percentge did the number of Sndy s mrbles increse? :06 Find the vlue of in ech cse. cm cm b 1 :0, 1:08 The probbility tht I will be llowed to leve before 10 m is 0. Wht is the probbility tht I won t be llowed to leve before 10 m? Clculte the re of ech of these shpes. b 6 cm c 6 8 cm Epnd 6( ). b Fctorise 1 0. c Epnd ( + ). d Fctorise 6b + 1b. 6 Convert 17 000 t to kg. b Chnge 7800 m to km. c Write 800 min in hours nd minutes. 7 Simplify: - b 1 6 8 c - d ( + 6) - 8 If 0% of fertiliser is superphosphte, how mny kilogrms of superphosphte re needed for 1 tonne of fertiliser? b A rel estte gent receives 1 % commission on the vlue of ech house tht she sells. How much does she receive for the sle of house vlued t $ 000? 9 Wht is the time difference from 1:0 on Tuesdy to 09:0 on Thursdy? b If I begn journey t 07: tht took 1 hours nd 0 minutes to complete, t wht time did I rrive t my destintion? 10 Clculte the volume of these solids, correct to two deciml plces. 1 cm b 6 cm 1 cm cm 8 cm cm 1 cm 7 cm 6:0 8:0 :08, :09 1:11 1:1 :06 8:08 8:06 9 cm 8 6 cm 10 cm 7 Austrlin Signpost Mthemtics New South Wles 8