Simultaneous. linear equations. and inequations

Transcription

1 Simultaneous linear equations and inequations A Graphical solution of simultaneous linear equations B Solving simultaneous linear equations using substitution C Solving simultaneous linear equations using elimination D Problem solving using simultaneous linear equations E Solving linear inequations F Sketching linear inequations G Solving simultaneous linear inequations What do ou know? List what ou know about linear equations and linear inequations. Create a concept map to show our list. Share what ou know with a partner and then with a small group. As a class, create a large concept map that shows our class s knowledge of linear equations and linear inequations. Digital doc Hungr brain activit Chapter doc-5 opening question How could John decide which of the two concreting companies he should use Angelico s Concrete ($7 plus $ per m of concrete) and Bau Cementing ($ plus $5 per m of concrete)?

2 Are ou read? Tr the questions below. If ou have difficult with an of them, etra help can be obtained b completing the matching SkillSHEET. Either search for the SkillSHEET in our ebookplus or ask our teacher for a cop. Digital doc Skillsheet. doc-5 Digital doc Skillsheet. doc-5 Digital doc Skillsheet. doc-5 Substitution into a linear rule Substitute for in each of the following equations to calculate the value of. a = - b = - c = Solving linear equations that arise when finding - and -intercepts For each of the following equations, substitute: i = to find the corresponding value of ii = to find the corresponding value of a + = b - = 9 c = - Transposing linear equations to standard form Write the following equations in the form = m + c. a + = 8 b 8 - = c = Digital doc Skillsheet. doc-55 Measuring the rise and the run Find the gradient for each of the following straight lines. a b 5 c Digital doc Skillsheet.5 doc-5 Digital doc Skillsheet. doc-57 Digital doc Skillsheet.7 doc-58 Finding the gradient given two points 5 Find the gradient of the line passing through each of the following pairs of points. a (, ) and (, 7) b (, -) and (, ) c (, ) and (-, ) Graphing linear equations using the - and -intercept method Graph each line with the following equations using the - and -intercept method. a 5 - = b - = 5 c + = Checking whether a given point makes the inequation a true statement 7 For each of the following, use substitution to check if the given point makes the inequalit a true statement. a - < (5, ) b Ç 5 + (-, ) c - > -8 (, ) 9 Maths Quest for the Australian Curriculum

3 A Graphical solution of simultaneous linear equations Simultaneous linear equations An two linear graphs will meet at a point, unless the are parallel. At this point, the two equations simultaneousl share the same - and -coordinates. This point is referred to as the solution to the two simultaneous linear equations. Simultaneous equations can be solved graphicall or algebraicall. Graphical solution This method involves drawing the graph of each equation on the same set of aes. The intersection point is the simultaneous solution to the two equations. An accurate solution depends on drawing an accurate graph. Graph paper or graphing software can be used. Worked Eample Use the graph of the given simultaneous equations below to determine the point of intersection and, hence, the solution of the simultaneous equations. + = = = - + = 5 Think Write Write the equations and number them. + = [] = - [] Locate the point of intersection of the two lines. This gives the solution. Check the solution b substituting = and = into the given equations. Comment on the results obtained. Point of intersection (, ) Solution: = and = Check equation []: LHS = + RHS = = + () = LHS = RHS Check equation []: LHS = RHS = - = = () - = - = LHS = RHS In both cases LHS = RHS, therefore the solution set (, ) is correct. Chapter Simultaneous linear equations and inequations 9

4 It is alwas important to check the solution. Worked Eample For the following simultaneous equations, use substitution to check if the given pair of coordinates, (5, -), is a solution. - = 9 [] + = - [] Think Write Write the equations and number them. - = 9 [] + = - [] Check b substituting = 5 and = - into equation []. Check b substituting = 5 and = - into equation []. Check equation []: LHS = - RHS = 9 = (5) - (-) = 5 + = 9 LHS = RHS Check equation []: LHS = + RHS = - = (-) + 5 = = - LHS = RHS In both cases, LHS = RHS. Therefore, the solution set (5, -) is correct. In order to obtain an accurate solution to a pair of simultaneous equations it is important to draw an accurate graph. This is demonstrated in the eample below. Worked Eample Solve the following pair of simultaneous equations using a graphical method. + = + = Think Write the equations, one under the other and number them. Calculate the - and -intercepts for equation []. For the -intercept, substitute = into equation []. For the -intercept, substitute = into equation []. Write/draw + = [] + = [] Equation [] -intercept: when =, + = = The -intercept is at (, ). -intercept: when =, + = = The -intercept is at (, ). 9 Maths Quest for the Australian Curriculum

5 Calculate the - and -intercepts for equation []. For the -intercept, substitute = into equation []. Divide both sides b. For the -intercept, substitute = into equation []. Divide both sides b. Use graph paper to rule up a set of aes and label the -ais from to and the -ais from to. 5 Plot the - and -intercepts for each equation. Produce a graph of each equation b ruling a straight line through its intercepts. 7 Label each graph. Equation [] -intercept: when =, + = = = The -intercept is at (, ). -intercept: when =, + = = = 5 The -intercept is at (, 5). 5 (, ) + = = - 8 Locate the point of intersection of the lines. The point of intersection is (, ). 9 Check the solution b substituting = and = into each equation. Check []: LHS = + RHS = = + = LHS = RHS Check []: LHS = + RHS = = () + () = + = LHS = RHS State the solution. In both cases, LHS = RHS. Therefore, the solution set (, ) is correct. The solution is =, =. A CAS calculator can be used to obtain a graphical (as well as an algebraic) solution to simultaneous linear equations. Parallel lines It is possible for two simultaneous linear equations to have no solution. This occurs when the graphs of the two equations do not cross because the have the same gradient. In other words, the two graphs are parallel. Consider the following pair of simultaneous equations. - = 5 [] - = [] The can be graphed to show two parallel lines = 5 - = 5 Chapter Simultaneous linear equations and inequations 9

6 We can confirm that the two lines are in fact parallel b rearranging each equation into the form = m + c and checking the gradient of each. - = 5 [] - = = = - 5 gradient m = - = [] - = - - = - + = - gradient m = It is also possible for two simultaneous linear equations to have man solutions. This occurs when the two linear equations are, in fact the same equation, simpl epressed in a different form. For eample, = - 5 [] - = 5 [] Simplifing equation [] b dividing b gives - = 5. Rearranging it in the same form as equation [] gives = - 5. A word of caution here: Make sure that the signs are eactl the same in both equations. The will not represent the same equation if this is not the case. remember. When solving simultaneous equations graphicall, obtaining an accurate solution depends on drawing accurate graphs.. The solution to linear simultaneous equations is the point where their graphs intersect.. Lines that have the same gradient are parallel.. If the graphs of the two simultaneous equations are parallel lines, then the simultaneous equations have no solution, as the have no point of intersection. Eercise A Individual Pathwas Activit -A Investigating graphs of simultaneous equations doc-99 Activit -A- Graphing simultaneous equations doc-99 Graphical solution of simultaneous linear equations Fluenc We Use the graphs below of the given simultaneous equations to write the point of intersection and, hence, the solution of the simultaneous equations. a + = b + = - = - = - = - = = = 9 Maths Quest for the Australian Curriculum

7 Individual Pathwas Activit -A- Further graphing of simultaneous equations doc-99 c - = + = 8 + = = d + = + = + = = e - = - = f - = 5 + = = - = = 5 + = Digital doc SkillSHEET. doc-57 WE For the following simultaneous equations, use substitution to check if the given pair of coordinates is a solution. a (7, 5) + = + = 8 b (, 7) - = + = 7 c (9, ) + = 5 - = d (, 5) + = 7 + = 8 e (, -) = = -5 f (, -) - = + = g (, -) + = - = 8 h (5, ) - 5 = - + = i (-, -5) - = - - = j (-, ) - = - = 7 WE Solve each of the following pairs of simultaneous equations using a graphical method. a + = 5 + = 8 b + = + = 5 c + = - = d - = -8 + = - e + 5 = 5 + = f + = + = 9 g = + = + 8 h = 8 + = 7 i - = -5 + = j + = - = k + = 7 + = l + = 8 + = Chapter Simultaneous linear equations and inequations 95

8 Solve each of the following pairs of simultaneous equations. a = 8 - b = + = + = + 8 d = + e = - = + = - c = - = 5 f + = = - g = 7 = + 5 h = + = - Understanding 5 Using technolog, determine which of the following pairs of simultaneous equations have no solutions. Confirm b finding the gradient of each line. a = - - = b 5 - = - = c + = = 5 d = = 8 e + = 9 + = f = 5 - = g + = = h - = - = Reasoning Two straight lines intersect at the point (, -). One of the lines has a -intercept of 8. The second line is a mirror image of the first in the line =. Determine the equation of the second line. (Hint: Draw a graph of both lines) 7 At a well-known beach resort it is possible to hire a jet-ski b the hour in two different locations. On the Northern beach the cost is $ plus $ per hour, while on the Southern beach the cost is $8 plus $8 per hour. The jet-skis can be rented for up to 5 hours. a Write the rules relating cost to the length of rental. b On the same set of aes sketch a graph of cost (-ais) against length of rental (-ais) for 5 hours. reflection c For what rental times, if an, is the Northern beach rental cheaper than the Southern beach rental? Use our graph to justif our answer. d For what length of rental time are the two rental schemes identical? Use the graph and our rules to justif our answer. What do ou think is the major error made when solving simultaneous equations graphicall? b Solving simultaneous linear equations using substitution There are two algebraic methods which can be used to solve simultaneous equations. The are the substitution method and the elimination method. Substitution method This method is particularl useful when one (or both) of the equations is in a form where one of the two variables is the subject. This variable is then substituted into the other equation, producing a third equation with onl one variable. This third equation can then be used to determine the value of the variable. 9 Maths Quest for the Australian Curriculum

9 Worked Eample Solve the following simultaneous equations using the substitution method. = - and + = 9 Think Write Write the equations, one under the other and number them. Substitute the epression ( - ) for from equation [] into equation []. Solve for. (i) Epand the brackets on the LHS of the equation. (ii) Collect like terms. (iii) Add to both sides of the equation. (iv) Divide both sides b. Substitute the value of into an of the equations, sa [], to find the value of. = - [] + = 9 [] Substituting ( - ) into []: + ( - ) = = 9 [] - = 9 = = Substituting = into []: = () - = - = 5 5 Write our answer. Solution: =, = 5 or (, 5) Check the answer b substituting the point of intersection into equation []. Check: Substitute into + = 9. LHS = () + (5) RHS = 9 = 9 + = 9 As LHS = RHS, the solution is correct. In some cases, both equations ma be written with the same variable as the subject. The can then be made equal to each other. This produces a third equation with onl one variable. Worked Eample 5 Solve the following pair of simultaneous equations using the substitution method. = 5-8 and = - + Think Write the equations, one under the other and number them. Both equations are written with as the subject, so equate them. Solve for. (i) Add to both sides of the equation. (ii) Add 8 to both sides of the equation. (iii) Divide both sides of the equation b 8. Substitute the value of into either of the original equations, sa [], and solve for. Write = 5-8 [] = - + [] 5-8 = = 8 = = Substituting = into []: = 5() - 8 = 5-8 = 7 Chapter Simultaneous linear equations and inequations 97

10 5 Write our answer. Solution: =, = 7 or (, 7) Check the answer b substituting the point of intersection into equation []. Check: Substitute into = - +. LHS = = 7 RHS = - + = -() + = -9 + = 7 As LHS = RHS, the solution is correct. remember When using the substitution method to solve simultaneous equations:. choose the equation in which one of the variables is the subject. substitute this epression for the variable into the other equation and solve. substitute the value ou have found into the rearranged equation to solve for the other variable. check our solution. Eercise b Individual Pathwas Activit -B Learning substitution doc-99 Activit -B- Practising substitution doc-99 Activit -B- Trick substitution doc-995 Solving simultaneous linear equations using substitution Fluenc We Solve the following simultaneous equations using the substitution method. Check our solutions using technolog. a = = b + = = c + = 7 = - - d + = = - 5 e = = f + = 9 = - 5 g = = h = = i = = - j = = 8 k + = = 9 - l = = 5 We 5 Solve the following pairs of simultaneous equations using the substitution method. Check our solutions using technolog. a = - and = + b = + 8 and = 7 - c = - and = - d = - 9 and = -5 e = - - and = - 8 f = and = + g = - - and = + h = + and = - i =.5 and = Maths Quest for the Australian Curriculum

11 j =. and =. +. k = - and = l = - and = - - Understanding A small farm has sheep and chickens. There are twice as man chicken as sheep, and there are legs between the sheep and the chickens. How man chickens are there? reflection When would ou choose the substitution method in solving simultaneous equations? C Interactivit Simultaneous equations int-78 Solving simultaneous linear equations using elimination Elimination is best used when the two equations are given in the form a + b = k. The method involves combining the two equations so that one of the variables is eliminated. Addition or subtraction can be used to reduce the two equations with two variables into one equation with onl one variable. Worked Eample Solve the following pair of simultaneous equations using the elimination method. - - = -9 + = 7 Think Write Write the equations, one under the other and number them. Look for an addition or subtraction that will eliminate either or. Note: Adding equations [] and [] in order will eliminate. Substitute the value of into equation []. Note: = ma be substituted into either equation. 5 Solve for. (i) Subtract from both sides of the equation. (ii) Divide both sides of the equation b. - - = -9 [] + = 7 [] [] + []: ( + ) = = - - = - Solve for b dividing both sides of the equation b -. = Substituting = into []: + = 7 = = Answer the question. Solution: =, = or (, ) 7 Check the answer b substituting the point of intersection into equation [] since equation [] was used to find the value of. Check: Substitute into - - = -9. LHS = -() - () = - - = -9 RHS = -9 LHS = RHS, so the solution is correct. When the like terms do not have the same coefficient, multipl one or both equations b a constant so as to create the same coefficient. Chapter Simultaneous linear equations and inequations 99

12 Worked Eample 7 Solve the following pair of simultaneous equations using the elimination method. - 5 = 7 + = 5 Think Write Write the equations, one under the other and number them. Look for a single multiplication that will create the same coefficient of either or. Multipl equation [] b and call the new equation []. - 5 = 7 [] + = 5 [] [] ì : - = - [] Subtract equation [] from [] in order to eliminate. [] - []: - - ( + ) = = 9 = -9 Solve for b dividing both sides of the equation = b. 5 Substitute the value of into equation []. Substituting = into []: + () = = 5 Solve for. (i) Subtract 9 from both sides of the equation. (ii) Divide both sides of the equation b. = - = - 7 Write our answer. Solution: = -, = or (-, ) 8 Check the answer b substituting into equation []. Check: Substitute into - 5 = 7. LHS = (-) - 5() = = 7 RHS = 7 LHS = RHS, so the solution is correct. Note: In this eample, equation [] could have been multiplied b - (instead of b ), then the two equations added (instead of subtracted) to eliminate. Sometimes it is necessar to multipl both equations b a constant in order to achieve the same coefficient for one of the variables. Worked Eample 8 Solve the following pair of simultaneous equations using the elimination method. + 5 = 5 + = Think Write the equations, one under the other and number them. Decide which variable to eliminate, sa. Multipl equation [] b and call the new equation []. Multipl equation [] b 5 and call the new equation []. Write + 5 = [] 5 + = [] Eliminate. [] ì : + = [] [] ì 5: 5 + = [] Maths Quest for the Australian Curriculum

13 Subtract equation [] from [] in order to eliminate. [] - []: ( + ) = = - = - Substitute the value of into equation []. Substituting = - into []: (-) + 5 = + 5 = 5 Solve for. (i) Add to both sides of the equation (ii) Divide both sides of the equation b 5. 5 = 5 = Write our answer. Solution: = -, = or (-, ) 7 Check the answer b substituting the solution into equation []. Check: Substitute into 5 + =. LHS = 5(-) + () = + = RHS = LHS = RHS, so the solution is correct. Note: Equation [] could have been multiplied b - (instead of b ), then the two equations added (instead of subtracted) to eliminate. remember. Simultaneous equations of the form a + b = k can be solved b the elimination method b looking for an addition or subtraction of the equations that will eliminate one of the variables.. For like terms with the same coefficient but opposite signs, add the equations. For like terms with the same coefficient and the same sign, subtract the equations.. If the terms do not have the same coefficient, multipl one or both equations b a constant to create the same coefficient. Remember to multipl both sides of the equation to keep it balanced.. Once one variable has been eliminated, solve the single variable equation formed. Substitute the solution back into one of the original equations to find the value of the variable that was originall eliminated. 5. Check our solution b substitution. Eercise C Individual Pathwas Activit -C Elimination practice doc-99 Solving simultaneous linear equations using elimination Fluenc We Solve the following pairs of simultaneous equations b adding equations to eliminate either or. a + = = b 5 + = 5 - = - c - + = + = Solve the following pairs of equations b subtracting equations to eliminate either or. a + = 5 + = b - 5 = + = 7 c - - = = Chapter Simultaneous linear equations and inequations

14 Individual Pathwas Activit -C- Let s eliminate doc-997 Activit -C- More elimination doc-998 Solve each of the following equations using the elimination method. a + = b + = - - = 5 + = -9 c + 5 = = -9 e - = 7 - = 7 g -5 + = -5 + = - i - - = = d - 5 = - - = - f - + = - = h 5-5 = - 5 = -5 WE7 Solve the following pairs of simultaneous equations. a + = b + = 9 - = - 5 = -7 c - + = 5 - = - e + = + = g = - + = - i - + = + = -9 k + = 9 + = -7 d + = = f 5 + = 7 + = h + 5 = + = -5 j - 5 = 7 + = l = = 9 5 WE8 Solve the following pairs of simultaneous equations. a + = b 5 - = + = 9 - = c + = + = e - = - 5 = g = -9 + = i - = = 7 k + = + = d + 7 = + = f = - + = h + 5 = - + = j - = - 5 = 9 l + = + = 5 Solve the following simultaneous equations using an appropriate method. Check our answer using technolog. a 7 + = b + = 8 = - + = c - + = = e = -7 = - d = 9 - = 7 f = - = Maths Quest for the Australian Curriculum

15 Digital doc WorkSHEET. doc-5 D Reasoning 7 Ann, Beth and Celine wanted to weigh themselves on a coin weighing machine. The problem was the onl had enough mone for one weighing. The decided to weigh themselves in pairs, one stepping off as another stepped on. Ann and Beth weighed 9 kg Beth and Celine weighed kg Celine and Ann weighed 5 kg How much did each of the girls weigh? reflection How does eliminating one variable help to solve simultaneous equations? Problem solving using simultaneous linear equations Man word problems can be solved using simultaneous linear equations. Follow these steps. Define the unknown quantities using appropriate pronumerals. Use the information given in the problem to form two equations in terms of these pronumerals. Solve these equations using an appropriate method. Write the solution in words. Check the solution. Worked Eample 9 Ashle received better results for his Maths test than for his English test. If the sum of the two marks is and the difference is, calculate the mark he received for each subject. Think Write Define the two variables. Let = the maths mark. Let = the English mark. Formulate two equations from the information given and number them. Note: Sum means to add and difference means to subtract. Use the elimination method b adding equations [] and [] to eliminate. + = [] - = [] [] + []: = 8 Solve for b dividing both sides of the equation b. = 9 5 Substitute the value of into equation []. Substituting = 9 into []: + = 9 + = Solve for b subtracting 9 from both sides of the = 7 equation. 7 Answer the question. Solution: Maths mark () = 9 English mark () = 7 8 Check the answer b substituting = 9 and = 7 into equation []. Check: Substitute into + =. LHS = RHS = = As LHS = RHS, the solution is correct. Chapter Simultaneous linear equations and inequations

16 Worked Eample To finish a project, Genevieve bus a total of 5 nuts and bolts from a hardware store. If each nut costs cents, each bolt costs 5 cents and the total purchase price is $., how man nuts and how man bolts does Genevieve bu? Think Write Define the two variables. Let = the number of nuts. Let = the number of bolts. Formulate two equations from the information given and number them. Note: The total number of nuts and bolts is 5. Each nut cost cents, each bolt cost 5 cents and the total cost is cents ($.). Solve simultaneousl using the substitution method since equation [] is eas to rearrange. Rearrange equation [] to make the subject b subtracting from both sides of equation []. 5 Substitute the epression (5 - ) for into equation []. + = 5 [] + 5 = [] Rearrange equation []: + = 5 = 5 - Substituting (5 - ) into []: (5 - ) + 5 = Solve for = + = + = = = 7 Substitute the value of into the rearranged equation = 5 - from step. Substituting = into = 5 - : = 5 - = 5 8 Answer the question. Solution: The number of nuts () = 5. The number of bolts () =. 9 Check the answer b substituting = 5 and = into equation []. Check: Substitute into + = 5. LHS = 5 + RHS = 5 = 5 As LHS = RHS, the solution is correct. remember. To solve worded problems, read the question carefull and define the two variables using appropriate pronumerals.. Formulate two equations from the information given and number them.. Use either the elimination method or the substitution method to solve the two equations simultaneousl.. Check our answer b substituting the values obtained for each variable into the original equations. Maths Quest for the Australian Curriculum

17 Eercise d Individual Pathwas Activit -D Problem solving doc-999 Activit -D- Harder problem solving doc-5 Activit -D- Trick problem solving doc-5 Problem solving using simultaneous linear equations Fluenc We 9 Rick received better results for his Maths test than for his English test. If the sum of his two marks is and the difference is, find the mark for each subject. We Rachael bus some nuts and bolts to finish a project. She does not bu the same number of nuts and bolts, but bus items in total. If each nut costs cents, each bolt costs cents and the total purchase price is $., how man nuts and how man bolts does she bu? understanding Find two numbers whose difference is 5 and whose sum is. The difference between two numbers is. If three times the larger number minus double the smaller number is, find the two numbers. 5 One number is 9 less than three times a second number. If the first number plus twice the second number is, find the two numbers. A rectangular house has a perimeter of metres and the length is metres more than the width. What are the dimensions of the house? 7 Mike has 5 lemons and oranges in his shopping basket. The cost of the fruit is $.5. Voula, with lemons and oranges, pas $. for her fruit. How much does each tpe of fruit cost? 8 A surveor measuring the dimensions of a block of land finds that the length of the block is three times the width. If the perimeter is metres, what are the dimensions of the block? 9 Julie has $. in change in her pocket. If she has onl 5 cent and cent pieces and the total number of coins is, how man coins of each tpe does she have? Mr Yang s son has a total of twent-one $ and $ coins in his monebo. When he counts his mone, he finds that its total value is $. How man coins of each tpe does he have? If three Magnums and two Paddlepops cost $8.7 and the difference in price between a Magnum and a Paddlepop is 9 cents, how much does each tpe of ice-cream cost? If one Redskin and Golden roughs cost $.5, whereas Redskins and Golden roughs cost $.55, how much does each tpe of sweet cost? Reasoning A catering firm works out its pricing based on a fied cost for overheads and a charge per person. It is known that a part for people costs $557, whereas a part for 5 people costs $99.5. Use this information to work out the fied cost and the cost per person charged b the compan. The difference between Sall s PE mark and Science mark is, and the sum of the marks is 5. If the PE mark is the higher mark, what did Sall get for each subject? Chapter Simultaneous linear equations and inequations 5

18 5 Mozza s cheese supplies sells si Mozzarella cheeses and eight Swiss cheeses to Munga s deli for $8., and four Mozzarella cheeses and four Swiss cheeses to Mina s deli for $8. How much does each tpe of cheese cost? If the perimeter of the triangle in the diagram is cm and the length of the rectangle is cm more than the width, find the value of and. cm Digital doc WorkSHEET. doc-5 E cm 5 cm ( + ) cm 7 Mr and Mrs Waugh want to use a caterer for a birthda part for their twin sons. The manager sas the cost for a famil of four would be $. However, the sons want to invite 8 friends, making people in all. The cost for this would be $. If the total cost in each case is made up of the same cost per person and the same fied cost, find the cost per person and the fied cost. 8 Joel needs to bu some blank DVDs and zip disks to back up a large amount of data that has been generated b an accounting firm. He bus DVDs and zip disks for $9. He later realises these are not sufficient and so bus another 5 DVDs and zip disks for $. How much did each DVD and each zip disk cost? (Assume the same rate per item was charged for each visit.) Solving linear inequations An equation is a statement of equalit such as = ; an inequation is a statement of inequalit such as < ( is less than ). The solution to a linear equation is a single point on a number line, but the solution to an inequation is a portion of the number line. That is, the solution to the inequation has man values. The following table shows four tpes of simple inequations and their corresponding representation on a number line. Note that an open circle placed over the indicates that is not included; that is, does not satisf the inequalit statement. A closed or solid circle indicates that is included; that is, it does satisf the inequalit statement. cm reflection How do ou decide which method to use when solving word problems using simultaneous linear equations? Mathematical statement English statement Number line diagram > is greater than í is greater than or equal to < is less than Ç is less than or equal to Maths Quest for the Australian Curriculum

19 The basic technique for solving inequations is to:. imagine that in place of the inequalit sign, there is an equals sign. solve the inequation as if it were a linear equation, ecept that in place of the equals sign keep the original inequalit sign unless the special case outlined below applies. Worked Eample Solve each of the following linear inequations. a + Ç b - < - c - 7 í + 5 Think Write a Write the inequation. a + Ç Obtain b subtracting from both sides of the inequation. Keep the inequalit sign the same throughout. + - Ç - Ç b Write the inequation. b - < - Add to both sides of the inequation. - + < - + < Obtain b dividing both sides of the inequation b. < < c Write the inequation. c - 7 í + 5 Combine the pronumeral terms b subtracting from both sides of the inequation í í 5 Add 7 to both sides of the inequation í í Obtain b dividing both sides of the inequation b. í í The special case multipling or dividing both sides of the inequation b a negative number Consider the inequation > 5 ( is greater than 5). If we multipl both sides of the inequation b we get: - > -5, which is not correct. In fact - < -5. Appling this to inequations generall, when we multipl or divide an inequation b a negative number, the direction of the inequalit sign must change. When multipling or dividing b a negative number, change the direction of the inequalit sign; that is, change: < to > > to < Ç to í í to Ç Chapter Simultaneous linear equations and inequations 7

20 Worked Eample Solve each of the following linear inequations. a -m + 5 < -7 b 5( - ) í 7( + ) Think Write a Write the inequation. a -m + 5 < -7 Subtract 5 from both sides of the inequation. (No change to the inequalit sign.) -m < m < Obtain m b dividing both sides of the m > inequation b -. Reverse the inequalit sign, since ou are dividing b a negative number. m > b Write the inequation. b 5( - ) í 7( + ) Epand both brackets. 5 - í 7 + Combine the pronumeral terms b subtracting 7 from both sides of the inequation í í Add to both sides of the inequation í + - í 5 Obtain b dividing both sides of the inequation b -. Since we need to divide b a negative number, reverse the direction of the inequalit sign. Ç Ç Ç 5 remember. The solution to an inequation is a portion of the number line. (That is, there are an infinite number of solutions to an given inequation.). When solving an inequation, imagine an equals sign in place of the inequalit sign and solve as if it was a linear equation. Remember to keep writing the original inequalit sign between the two sides of each step.. Special case: if in the process of the solution ou need to multipl or divide both sides of the inequation b a negative number, reverse the inequalit sign. That is, change < to >, > to <, Ç to í and í to Ç. Eercise e Individual Pathwas Activit -E Puzzling inequations doc-5 Activit -E- Puzzling inequations doc-5 Solving linear inequations fluenc We a Solve each of the following inequations. a + > b a + > c - í d m - í e p + < 5 f + < 9 g m - 5 Ç h a - Ç 5 i - > j 5 + m í 7 k + q í l 5 + a > - Solve each of the following inequations. Check our solutions b substitution. a m > 9 b 5p Ç c a < 8 d í e 5p > -5 f Ç 8 Maths Quest for the Australian Curriculum

21 Individual Pathwas Activit -E- Puzzling inequations doc-5 Digital doc Skillsheet.7 doc-58 m g m í h b > - i > j < k a 7 Ç - l m 5 í 5 We b Solve each of the following inequations. a m + < b + í c 5p - 9 > d n - Ç 7 e b - < f 8 - > g m + Ç - h a + 5 í -5 i b + < j c + 7 Ç k p - > l a - 7 í -8 We c Solve each of the following inequations. a m + > m + b a - í a c 5a - < a - 7 d a + Ç a - e 5 - > - f 7-5 Ç - g 7b + 5 < b + 5 h (a + ) > a + i (m - ) < m + j 5(m - ) Ç m + k (5b + ) Ç + b l 5(m + ) í (m + 9) 5 Solve each of the following inequations. a + Ç b + 7 í - c < 5 d > e í f < 7 We Solve each of the following inequations. a -m > b -5p Ç 5 c -a í d -p - Ç e - í f - < 7 g - p > h - a Ç i ( - ) < j -(a + 9) í 8 k 5 Ç -( + b) l - > 5 + m k + 5 < k - n ( - ) < 5( + 5) o 7(a + ) í (a - ) 7 MC When solving the inequation - > -7 we need to: A change the sign to í B change the sign to < C change the sign to = D change the sign to Ç E keep the sign unchanged 8 Solve each of the following inequations. a 5 m > b í 8a < e c < - 5 m Ç m + d Ç f 9 Solve each of the following inequations. a k > b -a - 7 < - c 5 - m í d + > 9 e - Ç f 5 + d < 7p m g í - h Ç i > 5 j 5a - < a + 7 k p + Ç 7p - l ( + ) > - understanding Digital doc Skillsheet.8 doc-59 Write linear inequations for the following statements, using to represent the unknown. (Do not attempt to solve the equations.) a The product of 5 and a certain number is greater than. b When three is subtracted from a certain number the result is less than or equal to 5. c The sum of seven and three times a certain number is less than. Given the positive numbers a, b, c and d and the variable, there is the following relationship: -c < a + b < -d. a Find the possible range of values of if a =, b =, c = and d = b Rewrite this relationship in terms of onl ( b itself between the < signs). Chapter Simultaneous linear equations and inequations 9

22 REASoning f Two speed boats are racing along a section of Lake Quikalong. The speed limit along this section of the lake is 5 km/h. Ross is travelling km/h faster than Steven and together the are travelling at a speed greater than km/h. a Write an inequation and solve it to describe all possible speeds that Steven could be travelling at. b At Steve s lowest possible speed, is he over the speed limit? c The water police issue a warning to Ross for eceeding the speed limit on the lake. Show that the police were justified in issuing a warning to Ross. At the beginning of this chapter we looked at the decision about which of two companies John should use when pouring different volumes of concrete. Angelico s Concrete charges $7 plus $ per cubic metre of concrete. Bau Cementing charges $ plus $5 per cubic metre of concrete. a Write an algebraic equation for the cost of using Angelico s Concrete. b Write an algebraic equation for the cost of using Bau Cementing. c Write an inequation that, when solved, will tell ou the volume of concrete for which it is cheaper to use Angelico s Concrete. d For what volume of concrete will it be cheaper to use Bau Cementing? e For what volume of concrete will the cost be the same (if an)? Sketching linear inequations Linear inequations replace the equalit sign with an inequalit sign, namel, >, í, < and Ç. The graph of linear inequations is a half plane and it is related to the graph of the corresponding linear equation in that the line forms the boundar between the two half planes. Consider the linear inequation í +. There are man points (, ) that satisf this inequation. All the points that lie on the line = + satisf this inequation, as well as man other points on the Cartesian plane for which the -coordinate is greater than two more than the -coordinate. For eample, (, 5), (, ), (-, ) are some points that satisf this inequation. The graph of an inequation is a half plane; in this eample, it is the region of the Cartesian plane above the line. The region that is required has been shaded, but sometimes it is helpful to shade the region that isn t required. Now consider the inequation < +. All the coordinates that have a -coordinate less than more than their -coordinate will satisf this inequation. Some solution points would be (, ), (, ) and (, ). To sketch the graph of this inequation we need to shade a region of the Cartesian plane that is below the line. In this eample the points that lie on the line are not part of the solution to the inequation, so the line is dotted, indicating that it is not included in the solution. reflection What is similar and different when solving linear inequations to linear equations? (-, ) (-, ) (-, ) (-, ) (, 5) (, ) (, 5) (, ) Maths Quest for the Australian Curriculum

23 Summar If the inequation is of the form of > m + c, then the region above the line is shaded and the line is dotted, indicating that the points that lie on the line are not part of the solution. m > Required region m < Required region If the inequation is of the form of í m + c, then the region above the line is shaded and the line is solid, indicating that the points that lie on the line are part of the solution. m > Required region m < Required region If the inequation is of the form of < m + c, then the region below the line is shaded and the line is dotted, indicating that the points that lie on the line are not part of the solution. m > Required region m < Required region Chapter Simultaneous linear equations and inequations

24 If the inequation is of the form of Ç m + c, then the region below the line is shaded and the line is solid, indicating that the points that lie on the line are part of the solution. m > Required region m < Required region Worked Eample Sketch the half plane given b each of the following inequations. a í + b < - Think Write a Write the inequation. a í + Sketch the linear equation, showing the - and -intercepts. = + Let =, = -. To find the -intercepts, let =. Therefore (-, ) is the -intercept. To find the -intercept, let =. The -intercept can be read from the equation, -intercept is (, ). 5 Sketch the line labelling the - and -intercepts. (-, ) (, ) Since the inequation is of the form í m + c, then the region above the line is shaded and the line is solid. The region of the plane where the points alwas have the -coordinate greater than or equal to the -coordinate plus two will be above the line. Therefore the region that is required is above the line. Shade this region. (, ) (-, ) Required region Maths Quest for the Australian Curriculum

25 b Write the inequation. b < - Sketch the linear equation, showing = - the - and -intercepts. To find the -intercepts, let =. Let =, =. To find the -intercept, let =. Therefore (, ) is the -intercept. The -intercept can be read from the equation. The -intercept is (, ). 5 Sketch the line labelling the - and -intercepts. Since the inequalit is of = the form < then this line should be a broken line. (, ) (, ) Since the inequation is of the form < m + c, then the region below the line is shaded and the line is dotted. < - (, ) (, ) Required region remember. Inequations involve the inequalit sign > (greater than), í (greater than and equal to), < (less than) and Ç (less than and equal to).. The graph of a linear inequation is a half plane.. A broken line is used for > or < signs, and a solid line is used for í and Ç signs.. If the inequation is of the form í m + c or > m + c, then the region above the line is the required region. 5. If the inequation is of the form Ç m + c or < m + c, then the region below the line is the required region. Eercise F Individual Pathwas Activit -F Understanding linear inequations doc-55 Sketching linear inequations fluenc We Sketch the half plane given b each of the following inequations. a í + b < - c > - - d < - e > - f < g í -5 h Ç - 7 i í - j < + 7 k < l Ç Chapter Simultaneous linear equations and inequations

26 Individual Pathwas Verif our solutions to question using technolog. MC a The shaded region satisfing the inequation > - is: A b c Activit -F- Graphing linear inequations doc-5 (, ) (, ) (, ) (, ) (, ) (, ) Activit -F- Interpreting linear inequation graphs doc-57 d e (, ) (, ) (, ) (-, ) b The shaded region satisfing the inequation Ç + is: A b (, ) (, ) (-, ) (-, ) c (, ) d (, ) (-, ) (-, ) e (, ) (, ) c The region satisfing the inequation < - is: A b (, ) (, ) (, ) (, -) Maths Quest for the Australian Curriculum

27 c (, ) d (, ) (, ) (, ) e (, ) (, ) understanding R l a Find the equation of the line l shown in the diagram at left. b Write down three inequations which define the region R. 5 Happ Yaps Dog Kennels charges $5 per da for large dogs (dogs over kg) and $ per da for small dogs (less than kg). On an da, Happ Yaps Kennels can onl accommodate a maimum of dogs. a If l represent the number of large dogs and s represents the number of small dogs. Write down an inequation, in terms of l and s, that represents the total number of dogs at Happ Yaps. b Another inequation can be written as s í. In the contet of this problem, write down what this inequation represents. c The inequation l Ç 5 represents the number of large dogs that Happ Yaps can accommodate on an da. This inequation is shown as a bold line on the graph below, clearl shade in the area that is not within the region for this inequation. l 5 s d Eplore the maimum number of small and large dogs Happ Yaps Kennels can accommodate to receive the maimum amount in fees. Chapter Simultaneous linear equations and inequations 5

28 Reasoning a Given the following graph, state the inequation it describes. b Prove, b choosing a point on the graph, that the inequation is correct. G reflection How are the graphs for linear equations and inequations similar and different? Solving simultaneous linear inequations The graph of a linear inequation represents a region of the Cartesian plane and not simpl a line. This means that two linear inequations will have two regions as graphs. If these regions intersect, the have an infinite number of points in common. It is easier to solve simultaneous inequations graphicall rather than algebraicall. The process involves drawing each half plane on a Cartesian plane, shading the regions that are required for each inequation. The region that is shaded b both inequations is the solution to the simultaneous inequations Worked Eample Use a graphical technique to solve the following simultaneous inequations. í + - > Think Write the inequations, one under the other and number them. Find the - and -intercepts for the boundar equation of inequation []. Rule a pair of coordinate aes and choose a suitable scale to allow all the intercepts to be marked. For inequation [] (í), sketch a solid line through both intercepts. 5 Label the inequation. For inequation [] test the point (, ) to see if a TRUE or FALSE statement is generated. Write/draw í + [] - > [] For [], the boundar is = +. -intercept: when =, = + + = = The -intercept is at (, ). -intercept: when =, = + = The -intercept is at (, ). Test point: (, ) Is í +? Is í? FALSE Maths Quest for the Australian Curriculum

29 7 As the statement is FALSE, the opposite side of the line is required. Shade the region required. Note: Since the equation is in the form í +, we would epect the required region to be above the line Region required Find the - and -intercepts for the boundar equation to inequation []. 9 For inequation [] (>), sketch a broken line through both intercepts. Label the inequation. For inequation [] test the point (, ) to see if a TRUE or FALSE statement is generated. As the statement is FALSE, the opposite side of the line is required. Shade the region required (below the dotted line). Indicate the solution region on the graph. This is the overlapping of the two shaded regions. Check our solution b substituting a point from the solution region, sa (7, 9), into each of the original inequations. For [], the boundar is - =. -intercept: when =, - = = = The -intercept is at (, ). -intercept: when =, () - = - = = - The -intercept is at (, -). Test point: (, ) Is () - >? Solution > - -8 FALSE Region required Check: Substituting (7, 9) into []: í + 9 í í 8 is true. Substituting (7, 9) into []: - > (7) - 9 > - 9 > 5 > is true. The solution region is correct. Chapter Simultaneous linear equations and inequations 7

30 remember. The graph of a linear inequation represents a region of the Cartesian plane.. A graphical technique can be used to solve simultaneous inequations.. The graph of two simultaneous inequations consists of the intersection of two regions and therefore has an infinite number of solutions.. It is usual to shade the wanted region on the graph of an inequation. Eercise G Individual Pathwas Activit -G Introducing simultaneous linear inequations doc-58 Activit -G- Practising simultaneous linear inequations doc-59 Activit -G- Further simultaneous linear inequations doc-5 Solving simultaneous linear inequations Fluenc Note: Questions and revise the skills used when working with inequations. For each of the following, use substitution to check if the given pair of coordinates makes the inequation true or false. a (, ) + > b (-, ) - < 7 c (, 5) Ç 5 + d (, -) 5 + < e (7, ) + 5 Ç 9 f (, ) - > g (-, -) - > h (-5, -) > 7 + i (, ) Ç - j (, ) + > Use the graphs of the equations given below to sketch the graph of the given inequations. (Remember to shade the region required.) a + > b + Ç - - c - > = = d + í = = -8 e í + f < - = = Maths Quest for the Australian Curriculum

31 g - < = h + í = 8 MC For each of the following pairs of simultaneous inequations, choose the graph which gives the correct solution. (Remember the required region is shaded.) a > + + Ç a 5-5 b 5-5 c 5 d E 5-5 b Ç 5 + > a b Chapter Simultaneous linear equations and inequations 9

32 c d E c í - + Ç a b c d E Maths Quest for the Australian Curriculum

33 d - < - í a b c d E understanding We Use a graphical technique to solve the following simultaneous inequations. a + < - í b + > + 5 Ç c > - < - d > + < - g + í + > 5 j - > + Ç m + > 7 - í 8 e - Ç 5 + > h > - < 5 k + > < n > í f < 8 + > 7 i - < í - l - í 9 + Ç 5 a Sketch the half plane represented b the region: i Ç + i i í -. b Show the region where both the inequations Ç + and í - hold true. Show the region where the inequations + < and - > simultaneousl hold true. Chapter Simultaneous linear equations and inequations

34 Digital doc WorkSHEET. doc-5 7 Natsuko is starting to plan a monthl budget b classifing ependitures such as rent and other epenses (r) and savings (). Her total net income is $ per month. She can spend no more than per cent of her income on rent. a Write an inequation to epress the constraint that Natsuko can spend no more than $ per month. b Write an inequation to epress the per cent rent and other epenses limitation. c Do an other inequations appl to this situation? Eplain. d Sketch a graph of the region that applies for all our inequations. e State three possible solutions of allocating rent and other epenses/savings. 8 Monica wants to take a minimum of 5 units of vitamin C and units of vitamin E per da. Each brand A tablet provides units of vitamin C and 5 units of vitamin E, while each brand B tablet provides 75 units of vitamin C and 75 units of vitamin E. a Write an inequation which indicates the combination needed of each brand of vitamin tablet to meet the dail requirement of vitamin C. (Hint: Let a = the number of brand A tablets and b = the number of brand B tablets.) b Write an inequation which indicates the combination needed of each brand of vitamin tablet to meet the dail requirement of vitamin E. c Graph the two inequations and indicate the region which provides a solution to both the vitamin C and vitamin E requirements. d Recommend to Monica two different vitamin plans that fit the restrictions. 9 Maria is making some high-energ sweets using peanuts and chocolate chips. She wanted to make a maimum of g of the sweets, but wanted them to contain at least 8 g of carbohdrates. a Let the mass of peanuts be p and the mass of chocolate chips be c. Write an inequation to represent the constraint that the total mass must be at most g. b On a Cartesian plane, sketch the region defined b the inequation obtained in part a. (Hint: Consider onl the positive aes as the values of both p and c must be positive.) c The peanuts provide % of their mass in carbohdrates and the chocolate chips provide % of their mass in carbohdrates. Write an inequation that represents the constraint that the mass of carbohdrates must be at least 8 g. d On a Cartesian plane, sketch the region defined b the inequation obtained in part c. e On a Cartesian plane, show the region where the inequations sketched in parts b and d both hold true. f The region obtained in part e shows all possible masses of peanuts and chocolate chips that meet Maria s requirements. List five sets of possible masses of peanuts and chocolate chips that would meet her requirements. reflection How do the solutions from a sstem of equations differ from a sstem of inequations? Maths Quest for the Australian Curriculum

35 Summar number and algebra Linear and non-linear relationships Graphical solution of simultaneous linear equations When solving simultaneous equations graphicall, obtaining an accurate solution depends on drawing accurate graphs. The solution to linear simultaneous equations is the point where their graphs intersect. Lines that have the same gradient are parallel. If the graphs of the two simultaneous equations are parallel lines, then the simultaneous equations have no solution, as the have no point of intersection. Solving simultaneous linear equations using substitution When using the substitution method to solve simultaneous equations: choose the equation in which one of the variables is the subject substitute this epression for the variable into the other equation and solve substitute the value ou have found into the rearranged equation to solve for the other variable check our solution. Solving simultaneous linear equations using elimination Simultaneous equations of the form a + b = k can be solved b the elimination method b looking for an addition or subtraction of the equations that will eliminate one of the variables. For like terms with the same coefficient but opposite signs, add the equations. For like terms with the same coefficient and the same sign, subtract the equations. If the terms do not have the same coefficient, multipl one or both equations b a constant to create the same coefficient. Remember to multipl both sides of the equation to keep it balanced. Once one variable has been eliminated, solve the single variable equation formed. Substitute the solution back into one of the original equations to find the value of the variable that was originall eliminated. Check our solution b substitution. Problem solving using simultaneous linear equations To solve worded problems, read the question carefull and define the two variables using appropriate pronumerals. Formulate two equations from the information given and number them. Use either the elimination method or the substitution method to solve the two equations simultaneousl. Check our answer b substituting the values obtained for each variable into the original equations. Solving linear inequations The solution to an inequation is a portion of the number line. (That is, there are an infinite number of solutions to an given inequation.) When solving an inequation, imagine an equals sign in place of the inequalit sign and solve as if it was a linear equation. Remember to keep writing the original inequalit sign between the two sides of each step unless the special case applies. Special case: if in the process of the solution ou need to multipl or divide both sides of the inequation b a negative number, reverse the inequalit sign. That is, change < to >, > to <, Ç to í and í to Ç. Chapter Simultaneous linear equations and inequations