number AnD AlgebrA Linear and non-linear relationships 11 linear equations 11A Identifying patterns 11B Backtracking and inverse operations 11C Keeping equations balanced 11d Using algebra to solve problems 11E Equations with the unknown on both sides WhAT Do you know? 1 List what you know about equations. Create a concept map to show your list. Share what you know with a partner and then with a small group. As a class, create a large concept map that shows your class s knowledge equations. Hungry brain activity Chapter 11 doc-6977 opening QuesTion A plane uses an average of 0 L of fuel an hour. Write an equation that will enable a pilot to input the speed and output the number of hours of flight.
number AnD AlgebrA Linear and non-linear relationships 1 a Add to obtain the net number in the sequence. The net three numbers are 1, 17, 19. b Subtract to obtain the net number in the sequence. The net Are you ready? three numbers are 1, 8,. Try the questions below. If you have difficulty with any of them, etra help can be obtained by completing the matching SkillSHEET located on your ebookplus. SkillSHEET 11.1 doc-6978 Number patterns 1 For each of the following sequences of numbers, describe the pattern in words and then write down the net three numbers in the pattern. a 7, 9, 11, 1, b 8,, 0, 16, c, 6, 1,, d 100000, 10000, 1000, SkillSHEET 11. doc-6979 SkillSHEET 11. doc-6980 Using tables to show number patterns a Complete the table shown, using the diagrams at right as a guide. Number of squares 1 6 Number of sides 8 1 16 0 b How many sides are there for 10 squares? 0 Describing a number pattern from a table Describe each number pattern shown in the tables. 1 square squares squares Each first number is two more than the previous first number and each second number is si more than the previous second number. Each second number is three times the matching first number. a First number 1 b First number 1 7 9 Second number 7 8 9 10 11 Second number 9 1 1 7 Each first number is one more than the previous first number and each second number is one more than the previous second number. Each second number is always 6 more than the matching first number. SkillSHEET 11. doc-6981 Flowcharts Complete these flowcharts to find the output number. a 11 ì -9 b + ó SkillSHEET 11. doc-698 Inverse operations Write the inverse operation for each of the following. a ì ó b + 8-8 c - 17 + 17 d ó - ì - SkillSHEET 11.6 doc-698 Solving equations by backtracking 6 Solve each of the following equations by first completing the flowcharts below. Remember to show the operations needed to backtrack to. + 9 a 7( - ) = b = - ì 7-7( - ) - 7( - ) + 9 + 9 9 + ó 7 Solution is = 9. 6 + 9 ó + 9 + 9 1-9 ì Solution is = 6. 9 Maths Quest 8 for the Australian Curriculum
number AnD AlgebrA Linear and non-linear relationships SkillSHEET 11.7 doc-698 Combining like terms 7 Simplify each of the following epressions by combining like terms. a 7v + + v + 10v + 7 b 6c + 1 - c - 8 c + 7 SkillSHEET 11.8 doc-698 Epanding epressions containing brackets 8 Epand each of the following epressions containing brackets. a ( + ) 6 + 10 b -7(m - 1) -7m + 7 SkillSHEET 11.9 doc-6986 Checking solutions by substitution 9 For each equation below there is a solution given. Is the solution correct? + 9 a - 7 = + = Yes b = 7 = No SkillSHEET 11.10 doc-6987 Writing equations from worded statements 10 Write an equation for each of the following statements, using to represent the unknown number. a When is added to a certain number, the result is 9. + = 9 b Eight times a certain number is 0. 8 = 0 c When 11 is subtracted from a certain number, the result is. - 11 = d Dividing a certain number by 6 gives a result of. 6 = Chapter 11 linear equations 9
number AND algebra Linear and non-linear relationships 11A Identifying patterns Mathematics is used to describe relationships in the world around us. Mathematicians study patterns in numbers and shapes found in nature to discover rules. These rules can then be applied to other, more general situations. Looking at the number pattern 1,, 7, 10,... we can see that by adding to any of these numbers, we obtain the net number. This number pattern is called a sequence. Each number in the sequence is called a term. Each sequence has a rule that describes the pattern. For the sequence above, the rule is add. Worked Eample 1 Describe the following number patterns in words then write down the net three numbers in the pattern. a, 8, 1,... b, 8, 16,... Think Write a 1 The net number is found by adding to the a Net number = previous number + previous number. Add each time to get the net three numbers. Write down the net three numbers. Net three numbers are 16, 0,. b 1 The net number is found by multiplying the b Net number = previous number ì previous number by. Multiply by each time to get the net three numbers. Write the net three numbers. Net three numbers are, 6, 18. Worked Eample Using the following rules, write down the first five terms of the number pattern. a Start with and divide by each time. b Start with, multiply by and subtract each time. Think Write a 1 Start with and divide by. a ó = 16 Keep dividing the previous answer by until five numbers have been calculated. 16 ó = 8 8 ó = ó = ó = 1 Write the answer. The first five numbers are 16, 8,, and 1. b 1 Start with then multiply by and subtract. b ì - = ì - = 17 Continue to apply this rule to the answer until five numbers have been calculated. 17 ì - = 6 6 ì - = 7 7 ì - = 10 Write the five numbers. The first five numbers are, 17, 6, 7, 10. 9 Maths Quest 8 for the Australian Curriculum
number AND algebra Linear and non-linear relationships Worked Eample Describe the pattern that occurs in the final digit of the number set represented by: 7 1, 7, 7, Think Write 1 Calculate a few of these powers. 7 1 = 7 Continue until a pattern in the last digit is noticed. When the number becomes large, be concerned with only the last digit. 7 = 9 7 = 9 ì 7 = 7 = ì 7 = 01 7 = 01 ì 7 = 16 807 7 6 =... 7 ì 7 =... 9 Write the pattern. The pattern in the last digit is 7, 9,, 1 repeated. Geometric patterns Patterns can be found in geometric shapes. If we eamine the three shapes below, we can see patterns by investigating the changes from one shape to the net. For eample, look at the number of matchsticks in each set of triangles. By using a table of values we can see a number pattern developing: Number of triangles Number of matchsticks 1 6 9 1 1 6 18 The pattern in the bottom row is, 6, 9,...; we can see that the rule here is add. We can also look for a relationship between the number of triangles and the number of matchsticks in each shape. If you eamine the table, you will see that a relationship can be found. In words, the relationship is the number of matchsticks equals times the number of triangles. Worked Eample Consider a set of heagons constructed according to the pattern shown below. a Using matches, pencils or similar objects, construct the above figures. Draw the net two figures in the series. b Draw up a table showing the relationship between the number of heagons in the figure and the number of matches used to construct the figure. Chapter 11 Linear equations 9
number AND algebra Linear and non-linear relationships c Devise a rule to describe the number of matches required for each figure in terms of the number of heagons in the figure. d Use your rule to determine the number of matches required to make a figure consisting of 0 heagons. Think a 1 Construct the given figures with matches. Note the number of additional matches it takes to progress from one figure to the net in this case. Write a The net two figures are Draw the net two figures, adding another matches each time. b Draw up a table showing the number of matches needed for each figure in terms of the number of heagons. Fill it in by looking at the figures. b Number of heagons 1 Number of matches 6 11 16 1 6 c 1 Look at the pattern in the number of matches going from one figure to the net. It is increasing by each time. If we take the number of heagons and multiply this number by, it does not give us the number of matches. However, if we add 1 to this number, it does give us the number of matches in each shape. d 1 Use the rule to find the number of matches to make a figure with 0 heagons. Work out the answer and write it down. c The number of matches increased by in going from one figure to the net. Number of matches = number of heagons ì + 1 d Number of matches for 0 heagons = 0 ì + 1 = 101 So 101 matches would be required to construct a figure consisting of 0 heagons. remember 1. A number pattern has a rule that can describe the pattern.. Geometric patterns have a rule that can describe the pattern.. The rule can be used to make predictions. 96 Maths Quest 8 for the Australian Curriculum
eercise 11A individual pathways Activity 11-A-1 Identifying patterns doc-6 Activity 11-A- More patterns doc-7 Activity 11-A- Advanced patterns doc-8 a Even numbers/ multiples of b Cubes c Prime numbers d Fibonacci sequence e Factors of 1 f Multiples/ powers of ii Number of squares 1 Number of matches 7 10 1 16 iii Number of matches = Number of squares ì + 1 iv 61 number AnD AlgebrA Linear and non-linear relationships Identifying patterns fluency 1 We1 Copy the patterns below, describe the pattern in words and then write down the net three numbers in the pattern. a,, 6, 8, b, 8, 1, 18, c 7,, 1, 18, d 1,, 9, 7, e 18, 6,, 16, f 1,, 9, 16, Fill in the missing numbers in the following number patterns. a,, 9, 1,, b 8,,, 1, c, 8,,, d,, 1, 1, e 66, 77,, 99, 11 f 100,,, 8, 80, We Using the following rules, write down the first five terms of the number patterns. a Start with 1 and add each time. 1,, 9, 1, 17 a 6, 1, 18 b Start with and multiply by each time., 1,, 1, 0 b 10, 1, 16 c Start with 0 and take away 8 each time. 0,,, 6, 18 c 16, 6 d Start with 6 and divide by each time. 6,, 16, 8, d 9, 11, 17 e 88, 110 e Start with 1, multiply by and add each time. 1,, 10,, 6 f 9, 90, 7 f Start with 1, add then multiply by each time. 1, 6, 16, 6, 76 understanding Each of the following represents a special number set. What is common to the numbers in the set? a,, 6, 8, 10 b 1, 8, 7, 6 c,,, 7, 11 d 1, 1,,,, 8 e 1,,,, 6, 1 f, 9, 7, 81 We Investigate the pattern that occurs in the final digit of the following sets. Describe the pattern in each case. a 1,,,,, 6, 8 b 1,,,, 9, 7, 1 c 1,,,, 6 d 1,,, e 8 1, 8, 8, 8,,, 6 6 We For each of the sets of shapes below: i Construct the shapes using matches. Draw the net two shapes in the series. ii Construct a table to show the relationship between the number of shapes in each figure and the number of matchsticks used to construct it. iii Devise a rule in words that describes the pattern relating the number of shapes in each figure and the number of matchsticks used to construct it. iv Use your rule to work out the number of matchsticks required to construct a figure made up of 0 such shapes. a 6 b i b 1 a Add ; 10, 1, 1 b Add ;, 8, c Subtract ; 1, 1, 9 d Multiply by ; 81,, 79 e Divide by ; 8,, f Squares of numbers 1;, 6, 9 iii Number of matches = Number of triangles ì + 1 iv 1 c i c d iii Number of matches = Number of houses ì + 1 iv 101 d i iii Number of matches = Number of fence units ì + 1 iv 61 6 a i Chapter 11 linear equations 97
number AND algebra Linear and non-linear relationships reasoning 7 Consider the triangular pattern of even numbers shown below. 7 a, b 6 8 10 1 1 16 18 0 6 8 0 6 6 8 0 6 8 0 6 8 10 1 8 60 Check with your teacher. 11B a Complete the net three lines of the triangle using this pattern. b Complete the triangle as far as necessary to find the position of the number 60. c Eplain how, without completing any more of the triangle, you could find the position of the number 100. Check with your teacher. d Study the triangle you have created in part and write down as many patterns as you can find. Illustrate each pattern with numbers from the triangle. Check with your teacher. e Create a similar triangle using odd numbers. Look for the patterns in this triangle. Are reflection they the same as or different from those for What should you look for when trying the triangle of even numbers? Justify your to determine number patterns? answers by illustrations from the triangle. Backtracking and inverse operations An equation links two epressions with an equals sign. Adding and subtracting are inverse operations. Multiplying and dividing are inverse operations. A flowchart can be used to represent a series of operations. In a flowchart, the starting number is called the input number and the final number is called the output number. By using inverse operations, it is possible to reverse the flowchart and work from the output number to the input number. Worked Eample Find the input number for this flowchart. - 7 ó - + 7 Think Write 1 Copy the flowchart. - 7 ó - + 7 Backtrack to find the input number. The inverse operation of + is - (7 - = ). The inverse operation of + - is ì - ( ì - = -8). The inverse operation of -7 is +7 (-8 + 7 = -1). Fill in the missing numbers. - 7 ó - + -1-8 + 7 ì - - 7 98 Maths Quest 8 for the Australian Curriculum
number AND algebra Linear and non-linear relationships Worked Eample 6 Find the output epression for this flowchart. ì + ó Think Write 1 Copy the flowchart and look at the operations that have been performed. ì + ó Multiplying by gives. ì + ó Adding gives +. ì + ó + Now place a line beneath all of + and divide by. ì + ó + + Worked Eample 7 Starting with, draw the flowchart whose output number is given by the epressions: a 6 - b -( + 6). Think a 1 Rearrange the epression. Note: 6 - is the same as - + 6. Multiply by -, and then add 6. b 1 The epression + 6 is grouped in a pair of brackets, so we must obtain this part first. Therefore, add 6 to. Write a ì - ó 6 - - + 6 b ó 6 ì - + 6 -( + 6) Multiply the whole epression by. remember 1. To work backwards through a flowchart we use inverse operations.. Adding and subtracting are inverse operations.. Multiplying and dividing are inverse operations. Chapter 11 Linear equations 99
number AnD AlgebrA Linear and non-linear relationships eercise 11b individual pathways Activity 11-B-1 Sudoku challenge A doc-9 Activity 11-B- Sudoku challenge B doc-0 Activity 11-B- Sudoku challenge C doc-1 backtracking and inverse operations fluency 1 We Find the input number for each of the following flowcharts. + 6 ì ó + a 8 b 0 8 7 ì + - ó - c - d -19 1 6 ì - - 6 ì - + ì ó -8 e -1 f -1 1-1 g + 11 ó - - - ó + 7 ì - - h - 1 i - 8 ó 6 ì - 0-7 ì - 8 j -11 k + 0. ì -.1 1. ó ì 1.07 l. We6 Find the output epression for each of the following flowcharts. ì - 7 7 ì a - 7 b (w - 7) w c s ì - + + ì - -s + d n -(n + ) e m ó + 7 m + 7 f y + 7 ó y + 7 g z ì 6 - ó + ì - ó 6 z h d ( d + ) i e ì ó + 1 ì -1 + ì e + 1 j ( - ) k w - ì - ó 7 + 6 ì - - 11 ( w ) l -(z + 6) - 11 z 7 00 Maths Quest 8 for the Australian Curriculum
number AND algebra Linear and non-linear relationships a b + 7 ì + 7 ( + 7) - 8 ì - - 8 -( - 8) m v - ó 6-8 v 8 6 n m ì 8 - ì -7-7(8m - ) c d ì - 6 m m m - 6 ì - - 6 m -m -m - 6 o k ó 6 ì - + k + 6 p p ì - - 7 ó p 7 e f g h i j k - ó 8 - - 8 ó 8 - - 8 8 ì - + 11 - - + 11 ì -1 + 11 - - + 11 ì -1-1 - - - 1 ì - + - - + ì - 7 ó - 7-7 11C WE7 Starting with, draw the flowchart whose output number is: a ( + 7) b -( - 8) c m - 6 d -m - 6 e f 8 8 g - + 11 h - + 11 i - - 1 j - k 7 m + n 8 7 o 1 6 + 7 p 11 l - ì - ó - -( - ) -( - ) + ó 8 - m + + + - 8 8 Keeping equations balanced As an equation can be thought of as two epressions with an equals sign between them, an equation can be thought of as a balanced scale. The diagram at right represents the simple equation =. reflection l ( ) What do you need to be careful of when you are backtracking equations? ó - ì -7 n - -7 ( - ) 1 1 1 = If the amount of the left-hand side (LHS) is doubled, the scale will stay balanced provided that the amount on the right-hand side (RHS) is doubled. 1 1 1 1 1 1 = 6 Similarly, the scale will stay balanced if we add a quantity to both sides. The scales will remain balanced as long as we do the same to both sides. 1 1 1 1 1 1 1 1 1 1 + = 8 Chapter 11 Linear equations 01
number AnD AlgebrA Linear and non-linear relationships WorkeD eample 8 Starting with the equation =, write the new equation when we: a multiply both sides by b take 6 from both sides c divide both sides by. Think WriTe a 1 Write the equation. a = Multiply both sides by. ì = ì Simplify by removing the multiplication signs. = 16 Write numbers before variables. b 1 Write the equation. b = Subtract 6 from both sides. - 6 = - 6 Simplify. - 6 = - c 1 Write the equation. c = Dividing by a fraction is the same as ó = ó multiplying by its reciprocal. Multiply both sides by. ì = ì Simplify. = 0 = 10 remember 1. An equation links two epressions with an equals sign.. An equation is like a pair of balanced scales (or a seesaw). The scales (or seesaw) will remain balanced as long as we do the same to both sides. eercise 11C individual pathways Activity 11-C-1 Riddle A doc- Activity 11-C- Riddle B doc- Activity 11-C- Riddle C doc- Interactivity Balancing equations int-0077 keeping equations balanced fluency 1 We8 Starting with the equation = 6, write the new equation when we: a add to both sides + = 11 b multiply both sides by 7 7 = c take from both sides - = d divide both sides by e multiply both sides by - - = - f multiply both sides by -1 - = -6 g divide both sides by -1 - = -6 h take 9 from both sides - 9 = - i multiply both sides by j divide both sides by = = 9 k take from both sides. = 1 understanding a Write the equation that is represented by the diagram at right. = b Show what happens when you halve the amount on both sides. Write the new equation. = 1 1 1 1 = 0 Maths Quest 8 for the Australian Curriculum
number AnD AlgebrA Linear and non-linear relationships a Write the equation that is represented by the diagram at right. + = b Show what happens when you take three from both sides. Write the new equation. = 1 1 1 1 1 1 1 1 a Write the equation that is represented by the diagram at right. + 1 = 7 b Show what happens when you add three to both sides. Write the new equation. + = 10 1 1 1 1 1 1 1 1 a Write the equation that is represented by the diagram at right. + 1 = b Show what happens when you double the amount on each side. Write the new equation. + = 10 1 1 1 1 1 1 WorkSHEET 11.1 doc-1 11D 6 MC If we start with =, which of these equations is not true? A + = 7 B = 8 C - = -10 d = 1 E - = 7 MC If we start with =, which of these equations is not true? A = B - = -6 C - 6 = 0 d = E - = 8 MC If we start with = -6, which of these equations is not true? A - = 6 B = -1 C - 6 = 0 d + = - E - = -8 9 MC If we start with = 1, which of these equations is not true? A reflection How does using inverse operations d = E + = 17 keep the equations balanced? using algebra to solve problems A linear equation is an equation where the variable has an inde (power) of 1. This means that it never contains terms like or. To solve a linear equation, perform the same operations on both sides until the variable or unknown is left by itself. Sometimes the variable or unknown is called a pronumeral. A flowchart is useful to show you the order of operations applied to, so that the reverse order and inverse operation can be used to solve the equation. As you become confident with solving equations algebraically, you can leave out the flowchart steps. Chapter 11 linear equations 0
number AND algebra Linear and non-linear relationships Worked Eample 9 Solve these one-step equations by doing the same to both sides. a p - = 11 b 16 = - Think Write a 1 Write the equation. a p - = 11 Draw a flowchart and fill in the arrow to show what has been done to p. - p p - Backtrack from 11. - p p - 16 11 + Add to both sides. p - + = 11 + Give the solution. p = 16 b 1 Write the equation. b 16 = - Draw a flowchart and fill in the arrow to show what has been done to. 16 16 Backtrack from -. ó 16 16 - - ì 16 Multiply both sides by 16. ì 16 = - ì 16 16 Give the solution. = - The equations in Worked eample 9 are called one-step equations because only one operation needs to be undone to obtain the value of the unknown. Worked Eample 10 Solve these two-step equations by doing the same to both sides. a ( + ) = 18 b + 1 = 7 Think Write a 1 Write the equation. a ( + ) = 18 0 Maths Quest 8 for the Australian Curriculum
number AND algebra Linear and non-linear relationships Draw a flowchart and fill in the arrow to show what has been done to. + ì + ( + ) Backtrack from 18. + ì + ( + ) 9 18 - ó Divide both sides by. ( + ) 18 = + = 9 Subtract from both sides. + - = 9-6 Give the solution. = b 1 Write the equation. b 1 7 Draw a flowchart and fill in the arrows to show what has been done to. ó + 1 + 1 Backtrack from 7. 18 ó + 1 + 1 6 7 ì - 1 Subtract 1 from both sides. + 1 1 = 7 1 = 6 Multiply both sides by. 6 6 Give the solution. = 18 The equations in Worked eample 10 are called two-step equations because two operations need to be undone to obtain the value of the unknown. Worked Eample 11 Solve the following equations by doing the same to both sides. a (m - ) + 8 = b 6 + 18 = Think Write a 1 Write the equation. a (m - ) + 8 = Chapter 11 Linear equations 0
number AND algebra Linear and non-linear relationships Draw a flowchart and fill in the arrows to show what has been done to m. - ì + 8 m m - (m -) (m - )+8 Backtrack from. - ì + 8 m m - (m - ) (m - )+8-1 - + ó - 8 Subtract 8 from both sides. (m - ) + 8-8 = - 8 (m - ) = - Divide both sides by. ( m ) = m = 1 6 Add to both sides. m - + = -1 + 7 Give the solution. m = b 1 Write the equation. b 6 + 18 = Draw a flowchart and fill in the arrows to show what has been done to. ó + ì 6 + 6( + ) Backtrack from -18. ó + ì 6 + 6( + ) -16-8 - -18 - ó 6 Divide both sides by 6. Subtract from both sides. 6 Multiply both sides by. 6 + 18 = 6 6 + = + = = 8 = 8 7 Give the solution. = -16 06 Maths Quest 8 for the Australian Curriculum
number AnD AlgebrA Linear and non-linear relationships eercise 11D individual pathways Activity 11-D-1 Using algebra to solve problems doc- Activity 11-D- More problems using algebra doc-6 Activity 11-D- Advanced problems using algebra doc-7 Spreadsheet -step equations doc- a m = b w = - c k = - d t = - e m = 1 f n = -18 g k = -9 h s = -7 i m =. j p = -1 k g = - l f = m q = 9.0 n r = -. o t =.6 p k = -0. q g = - 1 or -0. r f = 1 16 f j = 19 7 or 7 remember 1. A linear equation is an equation where the variable has an inde (power) of 1.. When solving linear equations, perform the same operations on both sides of the equation until the unknown is left by itself.. You can draw a flowchart to help you to decide what to do net. using algebra to solve problems fluency 1 We9a Solve these one-step equations by doing the same to both sides. a + 8 = 7 = -1 b 1 + r = 7 r = - c 1 = t + 7 t = d w +. = 6.9 w =.7 e = m + 1 m = 1 f = j + g q - 8 = 11 q = 19 h -16 + r = -7 r = 9 8 8 7 i 1 = t - 11 t = j y -.7 = 8.8 k - 11 = z - z = - 19 l - 9 = f - 1 f = 7 1 1 1 y = 1. We9b Solve these one-step equations by doing the same to both sides. a 11d = 88 d = 8 b 7p = -98 p = -1 c u = u = or 0.8 d.g = 1. g = e 8m = 1 f - = 9j g t 8 = h k m = 1 j = 1 t = = -1 k = -60 1 i -. = l v j 6 = c l = -1. v = k 9 = - l - 7 = h h = - or -11 7 1 c = - or 1 1-1 We 10a Solve these two-step equations by doing the same to both sides. a m + = 1 b -w + 6 = 16 c -k - 1 = 8 d t - = -1 e (m - ) = -6 f -(n + 1) = 18 g (k + 6) = -1 h -6(s + 11) = - i m + = 10 j 0 = -(p + 6) k - g = 1 l 11 - f = -9 m q -.9 = 1. n 7.6 + r = -8. o 1.6 = t - 0.8 p -6k + 7. = 8. q -g - 1 = r - 8 = f - 18 8 We 10b Solve these two-step equations by doing the same to both sides. a 9 m + h = 1 b = 1 = 9 c = 7 m = -17 d + = h = -1 e m w m c 7 = 1 m = -0 f = w = -10 g = 1 m = 1 h = 7 c = 1 i m t + c 1 = 10 m = -8 j = t = -7 k =. l 7 9 8. 8 c = -19. = -0.8 We 11 Solve these equations by doing the same to both sides. They will need more than two steps. ( + ) m + 6 a (m + ) + 7 = b = 6 c = m = 1 1 d = 1 m = - = -0 7 = e + b = 6 = -7 f = = 11 g = 1 = 7 h = 7 b = 10 i 7 f + = j 6 z = k 8 6 m u f = -9 z = 6 = m = l 9 = 9 11 u = -11 m m = 7 n -7(w + ) = o 6 10 = p d 7 m = - + 10 = 8 d = w = -1 = 8 n + 1 ( t ) q = n = 9 r + 9 = 6 t = - 7 Chapter 11 linear equations 07
number AnD AlgebrA Linear and non-linear relationships understanding Spreadsheet -step equations doc- 6 Below is Ale s working to solve the equation + = 1. + = 1 1 + = + = 7 Ale should have also divided by in the + - = 7 - second line or subtracted from both sides first = before dividing both a Is the solution correct? The solution is not correct. sides by. The solution b If not, can you find where the error is and correct it? should be = 1. 7 Simplify the left-hand side of the following equations by collecting like terms, and then solve. a + + + = 19 = b 1v - v + v = - v = - c -m + 6 - m + 1 = 1 m = -1 d -y + 7 + y - = 9 y = e -y - 7y + = 6 y = -6 f t + - 8t = 19 t = - g w + w - 7 + w = 1 w = h w + 7 + w - 1 + w + 1 = - w = 9 i 7 - u + + u = 1 u = - j 7c - - 11 + c - 7c + = 8 c = 6 0 could be the charge per hour; could be the flat fee covering travel and other epenses. reasoning 8 A repair person calculates his service fee using the equation F = 0t +, where F is the service fee in dollars and t is the number of hours spent on the job. a How long did a particular job take if the service fee was $1? 1 hours b Eplain what the numbers 0 and could represent as costs in the service fee equation. 9 Lyn and Peta together raised $17 from their cake stalls at the school fete. If Lyn raised l dollars and Peta raised $86, write an equation that represents the situation and determine the amount Lyn raised. l + 86 = 17, l = $1 10 a Write an equation that represents the perimeter of the figure at right and then solve for. 10 + = 18, = 1 cm + 18 + 19 + 17 Perimeter = 18 cm b Write an equation that represents the perimeter of the + 16 figure at right and then solve for. 11 + 1 = 87, = cm - 1 + 0 11 Two positive integers have a difference of and a sum of 1. Find the two numbers. 9 and 1 If four times a certain number equals nine minus a half of the number, find the number. 1 Tom is years old and his dad is 10 times his age, being 0 years old. Is it possible, at any stage, for Tom s dad to be twice the age of his son? Eplain your answer. 1 Laurie earns the same amount for mowing the four of the neighbour s lawns every month. Each month he saves all his pay ecept $0, which he spends on his mobile phone. If he has $600 at the end of the year, how much did he earn each month? (Write an equation to solve for this situation first.) $80 per month Yes, when Tom is and his dad is 90. 08 Maths Quest 8 for the Australian Curriculum - 1 Perimeter = 87 cm
number AND algebra Linear and non-linear relationships 1 The linear relationship between two variables and y is displayed in this table. y 1 8 7 1 1 6 8 1 1 1 Write the linear relationship as an equation. y = + 1 reflection How do you decide on the order to undo the operations? 11E Equations with the unknown on both sides Some equations have unknowns on both sides of the equation. If an equation has unknowns on both sides, eliminate the unknowns from one side and then solve as usual. Consider the equation + 1 = +. Drawing the equation on a pair of scales, looks like this: 1 1 1 1 1 1 + 1 = + The scales remain balanced if is eliminated from both sides: 1 1 1 1 1 1 Writing this algebraically, we have + 1 = + + 1 - = + - + 1 = + 1-1 = - 1 = = = + 1 = Worked Eample 1 Solve the equation t - 8 = t + 1 and check your solution. Think Write 1 Write the equation. t - 8 = t + 1 Subtract the smaller unknown (that is, t) from both sides and simplify. t - 8 - t = t + 1 - t t - 8 = 1 Chapter 11 Linear equations 09
number AND algebra Linear and non-linear relationships Add 8 to both sides and simplify. t - 8 + 8 = 1 + 8 t = 0 Divide both sides by and simplify. t 0 = t = 10 Check the solution by substituting If t = 10, t = 10 into the left-hand side and then the right-hand side of the equation. LHS = t - 8 = 0-8 = If t = 10, RHS = t + 1 = 0 + 1 = 6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when t = 10. Worked Eample 1 Solve the equation n + 11 = 6 - n and check your solution. Think Write 1 Write the equation. n + 11 = 6 - n The inverse of -n is +n. Therefore, add n to both sides and simplify. n + 11 + n = 6 - n + n n + 11 = 6 Subtract 11 from both sides and simplify. n + 11-11 = 6-11 n = - Divide both sides by and simplify. n = n = -1 Check the solution by substituting n = -1 into the left-hand side and then the right-hand side of the equation. If n = -1, LHS = n + 11 = - + 11 = 8 If n = -1, RHS = 6 - n = 6 - - = 6 + = 8 6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when n = -1. Worked Eample 1 Epand the brackets and then solve the following equation, checking your solution. a (s + ) = (s + 7) + b (d + ) - (d + 7) + = (d + ) + 7 10 Maths Quest 8 for the Australian Curriculum
number AND algebra Linear and non-linear relationships Think Write a 1 Write the equation. a (s + ) = (s + 7) + Epand the brackets on each side of the equation first and then simplify. Subtract the smaller unknown term (that is, s) from both sides and simplify. s + 6 = s + 1 + s + 6 = s + 18 s + 6 - s = s + 18 - s s + 6 = 18 Subtract 6 from both sides and simplify. s + 6-6 = 18-6 s = 1 Check the solution by substituting s = 1 into the left-hand side and then the right-hand side of the equation. If s = 1, LHS = (s + ) = (1 + ) = (1) = If s = 1, RHS = (s + 7) + = (1 + 7) + = (19) + = 8 + = 6 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when s = 1. b 1 Write the equation. b (d + ) - (d + 7) + = (d + ) + 7 Epand the brackets on each side of the equation first, and then simplify. Subtract the smaller unknown term (that is, d) from both sides and simplify. Rearrange the equation so that the unknown is on the left-hand side of the equation. d + 1 - d - 1 + = d + 10 + 7 d + = d + 17 d - d + = d - d + 17 = d + 17 d + 17 = Subtract 17 from both sides and simplify. d + 17-17 = - 17 d = -1 6 Divide both sides by and simplify. d 1 = d = - 7 Check the solution by substituting d = - into the left-hand side and then the right-hand side of the equation. If d = -, LHS = (d + ) - (d + 7) + = (- + ) - (- + 7) + = (-) - () + = -8 - + = -8 If d = -, RHS = (- + ) + 7 = (-) + 7 = -1 + 7 = -8 8 Comment on the answers obtained. Since the LHS and RHS are equal, the equation is true when d = -. Note: When solving equations with the unknown on both sides, it is good practice to remove the unknown with the smaller coefficient from the relevant side. Chapter 11 Linear equations 11
number AnD AlgebrA Linear and non-linear relationships remember 1. When a unknown appears on both sides of an equation, remove the unknown term from one side. It is good practice to remove the smaller unknown from the relevant side.. For a positive term we can remove by subtraction. For eample, 7 + 7 = - (subtract from both sides).. For a negative term we can remove by addition. For eample, + 11 = 7 - (add to both sides). eercise 11e individual pathways Activity 11-E-1 Rocket A doc-8 Activity 11-E- Rocket B doc-9 Activity 11-E- Rocket C doc-0 Spreadsheet Unknowns on both sides doc- equations with the unknown on both sides fluency 1 We 1 Solve the following equations and check your solutions. a 8 + = 6 + 11 = b y - = y + 7 y = c 11n - 1 = 6n + 19 n = d 6t + = t + 17 t = e w + 6 = w + 11 w = f y - = y + 9 y = g z - 1 = z - 11 z = h a + = a - 10 a = - i s + 9 = s + s = j k + = 7k - 19 k = k w + 9 = w + w = - l 7v + = v - 11 v = - We 1 Solve the following equations and check your solutions. a w + 1 = 11 - w w = b b + 7 = 1 - b b = c n - = 17-6n n = d s + 1 = 16 - s s = e a + 1 = -6 - a a = - f 7m + = -m + m = g p + 7 = -p + 1 p = h + d = 1 - d d = i + m = - m m = 0 j 7s + = 1 - s s = 1 k t - 7 = -17 - t t = - l 16 - = + = We 1 Epand the brackets and then solve the following equations, checking your solutions. a ( + 1) + = 0 = b (m - 7) + m = 76 m = 10 c (n - 1) = (n + ) + 1 n = 1 d t + = (t - 7) t = e d - = ( - d) d = f ( - w) = w + 1 w = 1 6 9 g (k + ) - (k - 1) = k - 7 k = 10 h ( - s) = -(s - 1) s = - i -(z + ) = ( - z) z = -17 j (v + ) = 7(v + 1) v = 1 1 k m + (m - 7) = + (m + ) m = 11 l d + (d + 1) = (d - 7) d = 7 10 m (d + ) (d + 7) + = (d + 1) d = 19 n (k + 11) + (k ) 7 = (k ) k = 10 o 7(v ) ( v) + = (v + ) 8 v = p (l 7) + (8 l) 7 = (l + ) 6 l = 18 understanding Solve the equation: ( + ) ( 1 ) = = - 1 7 Solve the equation: ( ) + = = 1 or 6 Find the value of given the perimeter of the rectangle is 8 cm. 6. cm + 1-6( + ) = 1 ì 6 ì ( + + ) 7 The two shapes below have the same area. a Write an equation to show that the cm parallelogram and the trapezium have the same area. 6 cm 6 cm b Solve the equation for. = 1 c State the dimensions of the shapes. ( + ) cm ( + ) cm Parallelogram: base cm, height 6 cm, Trapezium: base 6 cm, top cm, height 6 cm. 1 Maths Quest 8 for the Australian Curriculum
number AnD AlgebrA Linear and non-linear relationships WorkSHEET 11. doc- Weblink Solving equations Interactivity Solving equations int-7 reasoning 8 A maths class has equal numbers of boys and girls. Eight of the girls left early to play in a netball match. This left times as many boys in the class as girls. How many students are in the class? 9 Judy is thinking of a number. First she doubles it and adds. She realizes that if she multiplies it by and subtracts 1, she gets the same result. Find the number. 10 Mick s father was 8 years old when Mick was born. If his father is now times as old as Mick is, how old are they both now? Mick is 1, his father is. + 8 11 Given the following number line for a line segment PR, determine the length of PR? 16 P Q R 1 In 8 years time, Tess will be times as old as her age 8 years ago. How old is Tess now? 1 reflection Does it matter to which side of the equation the unknowns are moved? Chapter 11 linear equations 1
number AnD AlgebrA Linear and non-linear relationships summary Identifying patterns A number pattern has a rule that can describe the pattern. Geometric patterns have a rule that can describe the pattern. The rule can be used to make predictions. Backtracking and inverse operations To work backwards through a flowchart we use inverse operations. Adding and subtracting are inverse operations. Multiplying and dividing are inverse operations. Keeping equations balanced An equation links two epressions with an equals sign. An equation is like a pair of balanced scales (or a seesaw). The scales (or seesaw) will remain balanced as long as we do the same to both sides. Using algebra to solve problems A linear equation is an equation where the variable has an inde (power) of 1. When solving linear equations, perform the same operations on both sides of the equation until the unknown is left by itself. You can draw a flowchart to help you to decide what to do net. Equations with the unknown on both sides When an unknown appears on both sides of an equation, remove the unknown term from one side. It is good practice to remove the smaller unknown from the relevant side. For a positive term we can remove by subtraction. For eample, 7 + 7 = - (subtract from both sides). For a negative term we can remove by addition. For eample, + 11 = 7 - (add to both sides). Homework Book MAPPING YOUR UNdERSTANdING Using terms from the summary, and other terms if you wish, construct a concept map that illustrates your understanding of the key concepts covered in this chapter. Compare this concept map with the one that you created in What do you know? on page 91. Have you completed the two Homework sheets, the Rich task and the two Code puzzles in your Maths Quest 8 Homework Book? 1 Maths Quest 8 for the Australian Curriculum
Chapter review number AND algebra Linear and non-linear relationships ì -1 Fluency 1 Find the output number for each of these flowcharts. - 1 ì a ( - 1) b c d ó 8 ì + 8 ó ì - 7 + ó Draw the flowchart whose output number is given by the following epressions. a -(m + ) + ì - b m m + -(m + ) n ó + n n n + c m 7-7 ó - m m - 7 m - 7 m - 7 - d 7-1w a Write an equation that is represented by the diagram below. + = 8 + 7 w -1w 7-1w Key represents an unknown amount represents 1 b Show what happens when you take from both sides, and write the new equation. (See bottom of page) MC If we start with =, which of these equations is not true? A + = 7 B = 1 8 C - = -10 d = 1 E - = b + 8 7 + MC If we start with =, which of these equations is not true? A = B - = -6 C - 6 = 0 d = E - = 6 Solve these equations by doing the same to both sides. a z + 7 = 18 11 b - + b = -18 7 c - 8 9 = z - d 9t = 1 l e 8. 7 = f - 6 1 = h 8 1 7 9 -. - 8 9 or - 1 1 7 Solve these equations by doing the same to both sides. a v + = 18 v = b (s + 11) = s = - c d 7 = 10 d = 7 d -(r + ) - = r = -9 e y = 9 7 y = f = 8 Solve the following equations and check each solution. a k + 7 = k + 19 k = b s - 8 = s - 1 s = - c t - 11 = - t t = d + = - + 16 = 9 Epand the brackets first and then solve the following equations. a (v + ) - 7v = 1 v = b (m - ) + m = m + 8 m = = 6 Chapter 11 Linear equations 1
1 c 1 hour minutes d $: charge per half hour, $86: flat call-out fee. number AND algebra Linear and non-linear relationships Problem solving 1 Ray the electrician charges $80 for a call-out visit and then $6 per half hour. a Write an equation for his fees, where C is his cost and t is the number of 0-minute periods he spent on the job. C = 6t + 80 b How long did a particular job take if he charged $7.00? 1 1 hours c Ray s brother Roger is a plumber and uses the equation C = t + 86 to calculate his costs. He also charged $7 for one job. How long did Roger spend at this particular job? d Eplain what the number and 86 could mean. At the end of the term, Katie s teacher gave the class their average scores. They had done four tests for the term. Katie s average was 76%. She had a mark of 8% for Probability, 7% for Geometry and 91% for Measurement but had forgotten what she got for Algebra. Write an equation to show how Katie would work out her Algebra test score and then solve this equation. Katie scored 8% for her Algebra test. My three daughters were each born years apart. Their combined ages come to 6 years. What is the age of the eldest? Eldest is years old. A truck carrying 0 bags of cement weighs. tonnes. After delivering 1 bags of cement, the truck weighs.17 tonnes. How much would an empty truck weigh? Truck weighs.11 tonnes. You are times as old as your sister. In 8 years time you will be twice as old as your sister. What are your ages now? You are 16 and your sister is. 6 What whole number must be added to both the numerator and the denominator of a fraction of 1 to obtain an answer of? 7 You lend three friends a total of $. You lend the first friend dollars. To the second friend you lend $ more than you do for the first friend. To the third friend you lend three times as much as for the second friend. How much does each person receive? $, $10, $0 8 What is the greatest possible perimeter of a triangle with sides + 0, + 76, + 196, given that the triangle is isosceles? All sides are in mm. 8 mm 9 Penny was looking at the following phone plans to see which one would be the most economical for her phone habits. Penny s average phone conversation was approimately minutes long. 1. 1 cents connection fee and 0 cents per minute y = 0.1 + 0. 16 Maths Quest 8 for the Australian Curriculum y = 0. + 0.19 where in the time of the call and y is the cost of the call.. No connection fee and 0 cents per minute y = 0.. 0 cents connection fee and 19 cents per minute a Help Penny by writing an algebraic equation for each phone plan. b Use the equations to find the most economical phone plan for Penny if her conversations average: 1 minute, minutes, minutes c At what average talking time should you switch to Plan? 10 A rectangular vegetable patch is ( + ) metres long and ( ) metres wide. Its perimeter is 8 metres. What are the dimensions of the vegetable plot? m ì 7 m 11 Knitting involves following patterns, often with outstanding designs resulting. Consider a simple pattern to knit a triangular scarf. The pattern might read like this. Start with stitches. Increase by 1 stitch at both ends of each row. Continue in this manner until the scarf is as long as you would like. a If r represents the row number, and n the number of stitches in that row, write an equation showing the relationship between n and r. b Use your equation to calculate the number of rows you would have to knit to have 6 stitches. rows 1 Three buckets (one black, one white and one red) each hold a maimum of L. Different volumes of water (in whole numbers of litres) are poured into each one. Twice the amount of water in the black bucket is equal to half the amount of water in the red bucket. If the water in the black bucket was poured into the white bucket, the total amount would be the same as the amount of water in the red bucket. How much water is in each bucket? Eplain the reasoning you use to arrive at your answer. 1 While on holidays Amy hired a bicycle for $9 an hour and $ for the use of a helmet. Her brother, Ben, found a cheaper hire place, which charged $6 per hour; but the hire of the helmet was $ and he had to pay $ for insurance. Both places measured the time in 0-minute increments. They both ended up paying the same amount and were gone for the same number of hours. Construct a table of values to determine how long they were gone and how much it cost? (See net page) n = r + 1 1 L in the black bucket, L in the white bucket, L in the red bucket.
number AnD AlgebrA Linear and non-linear relationships 1 Michael checked his bank balance before going shopping. He had $0. While shopping he paid with his bank card. He bought two suits, which cost the same, and three pairs of shoes, each of which cost half the price of a suit. He also had lunch for $1. When he checked his balance again he was $ overdrawn. What did one suit cost? $1.7 1 Shannon is saving to buy a new computer, which costs $99. So far he has $9 in the bank and he wants to make regular deposits each month until he reaches his target of $99. If he wants to buy the computer in 8 months time, how much does he need to save as a monthly deposit? $6. 1 hours 0 minutes Time Amy Ben 0 $ $10 0 $6 $1 0 $9 $1 60 $1 $16 80 $1 $18 100 $18 $0 10 $1 $ 10 $ $ Interactivities Test yourself Chapter 11 int-7 Word search Chapter 11 int-6 Crossword Chapter 11 int-6 Chapter 11 linear equations 17
ebook plus ACTiviTies Chapter opener (page 91) Hungry brain activity Chapter 11 (doc-6977) Are you ready? s (pages 9 9) SkillSHEET 11.1 (doc-6978) Number patterns SkillSHEET 11. (doc-6979) Using tables to show number patterns SkillSHEET 11. (doc-6980) Describing a number pattern from a table SkillSHEET 11. (doc-6981) Flowcharts SkillSHEET 11. (doc-698) Inverse operations SkillSHEET 11.6 (doc-698) Solving equations by backtracking SkillSHEET 11.7 (doc-698) Combining like terms SkillSHEET 11.8 (doc-698) Epanding epressions containing brackets SkillSHEET 11.9 (doc-6986) Checking solutions by substitution SkillSHEET 11.10 (doc-6987) Writing equations from worded statements 11A Identifying patterns s (page 97) Activity 11-A-1 (doc-6) Identifying patterns Activity 11-A- (doc-7) More patterns Activity 11-A- (doc-8) Advanced patterns 11B Backtracking and inverse operations s (page 00) Activity 11-B-1 (doc-9) Sudoku challenge A Activity 11-B- (doc-0) Sudoku challenge B Activity 11-B- (doc-1) Sudoku challenge C 11d Using algebra to solve problems s (page 07) Activity 11-D-1 (doc-) Using algebra to solve problems Activity 11-D- (doc-6) More problems using algebra Activity 11-D- (doc-7) Advanced problems using algebra Spreadsheet (doc-) -step equations (page 07) Spreadsheet (doc-) -step equations (page 08) 11E Equations with the unknown on both sides s (page 1) Activity 11-E-1 (doc-8) Rocket A Activity 11-E- (doc-9) Rocket B Activity 11-E- (doc-0) Rocket C Spreadsheet (doc-) Unknowns on both sides WorkSHEET 11. (doc-) (page 1) Weblink (page 1) Solving equations Interactivity (page 1) Solving equations (int-7) Chapter review Interactivities (page 17) Test yourself Chapter 11 (int-7) Take the end-ofchapter test to test your progress. Word search Chapter 11 (int-6) Crossword Chapter 11 (int-6) To access ebookplus activities, log on to www.jacplus.com.au 11C Keeping equations balanced s (page 0) Activity 11-C-1 (doc-) Riddle A Activity 11-C- (doc-) Riddle B Activity 11-C- (doc-) Riddle C WorkSHEET 11.1 (doc-1) (page 0) Interactivity (page 0) Balancing equations (int-0077) 18 Maths Quest 8 for the Australian Curriculum